As I mentioned in a comment, Osbert Sitwell's novel The Man Who Lost Himself (1929) was referenced in the sf-encyclopedia.com article "Time Paradoxes" as an early story dealing with the idea of meeting one's older/younger self, and @user14111 also mentioned that the novel can be borrowed on the internet archive here. I went ahead and read it, I'd say it's a sort of ambiguous case since it might be said to fall into the literary tradition that Lovecraft labeled as the "weird tale", with a lot of mysterious and unexplained dream-like elements that make it unclear if this was physical time travel or something else, including some differences between the encounters described from both perspectives. (As noted at the end of the the June 1 2020 entry here, Lovecraft actually had the novel in his library and put it on his list of "Weird &c. Items in Library of H. P. Lovecraft".) But I'll describe the story with a focus on the details that seem most relevant to questions of whether this was time travel or a premonition or something else, you can judge for yourself whether it fits.
The story is written from the perspective of an unnamed narrator describing the life and death of a British writer, Tristram Orlander, who the narrator had once been friends with. They had known each other since their school days, and the narrator starts by describing his youth, at one point mentioning that Tristram seemed to have "an indubitable and curious power of intuition—or perhaps one should say, rather, of being aware of small happenings that would shortly take place without the necessity of being informed of them beforehand." (p. 13)
Then the narrator moves on to the main narrative which will take up most of the novel, with some earlier comments on p. 9-10 suggesting this main narrative takes place in the late 1920s and Tristram is around 28. At this point he is a bit of a "starving artist" type whose poems and novels are judged to be great by a small audience with discriminating tastes, but not by the wider public. Tristram has a nervous breakdown after being rejected by a woman he fell in love with, so his doctor suggests a trip through Europe to recuperate, and the narrator volunteers to accompany him. Eventually they arrive in the Spanish city of Grenada, which they appreciate for its old world charm, and stay there for a long time during which Tristram seems to be recovering his health.
At one point in this section, there is some more significant discussion of premonitions of the future when Tristram and the narrator meet up with some other writers they know who are staying at a large hotel called the Hotel Boabdil (Tristram and the narrator are not staying there, but at a much smaller hotel with only a few guests). The discussion had turned to more serious or metaphysical topics, including "the distorted mirroring of the future that sometime occurs—or perhaps only seems to us in days after to have occurred—in a dream; a distortion similar to that evinced in one that follows certain events upon which obviously it is based, except that in these cases the order is reversed, and the dream precedes the happenings upon which it is founded. FitzMaurice held that if this predictive distortion could be proved and studied, much was to be deduced from it as to the nature of Time." (p. 164) Tristram recalls his childhood ability to predict minor events, and one of the other writers suggests that in cases like this one is not merely passively predicting things, but also subconsciously causing them in some sense: "if you either wanted or dreaded something sufficiently, you exercised on it, through the concentration that took place, an unconscious influence, equivalent to that of a magnet upon a nail, which would surely, though without your knowledge, draw it toward you out of the future." (p. 166) Tristram expands on this idea with some examples he's seen and speculates that "perhaps, if you ask a question of destiny, you will receive an answer in keeping with it; but the answer is in reality only the echo which the question calls forth." (p. 169)
After a while the narrator has to return to London, but thinks that Tristram seems sufficiently recovered that he will be OK staying on his own in Granada for a while longer. But once he no longer has company, the symptoms of his breakdown begin to return, including depressive thoughts about wasting his life pursuing an artistic career that has gotten him so little recognition or money, and disturbing dreams which cause him insomnia. There are some local personalities in Grenada who he finds vaguely disturbing and who keep reappearing in these dreams: a French woman who has become a beggar and goes around asking for money with her malnourished-looking child following everywhere, and the concierge at the hotel Boabdil, who had reminded both Tristram and the narrator of a puppet with his stiff motions and tendency to pop up from behind the counter when people entered. On one night he has a dream where the concierge points to the Frenchwoman and her child and says "It is the child, sir, to whom I want to call your attention." (p. 198) The concierge then leads him to a disturbing encounter with the woman he had fallen in love with, whose face has become a skull, and when he turns around he sees that "the child all at once grew up, and became the concierge, dressed in his uniform".
The next morning he decides he would be better off with company, so he has the plan to ask the concierge Hotel Boabdil if a certain made-up person is staying there, and then when the concierge says no, demand to inspect the guest book so he can see if anyone he might know is there. But when he gets there he is in a nervous state and blanks on the name he was going to ask about, so instead he just decides to use his own name, saying "I've come to see Mr. Tristram Orlander. Is he in?" (p. 218) While saying this he has a strange sense of deja vu, and then to his surprise the concierge says "Yes sir, I believe he's waiting for you", and takes him to a room. He knocks on the door and is invited to come in, where he sees "an elderly, bearded stranger" (p. 219) in an expensively-furnished room who rises from his chair and says "I was expecting you, but not yet: you, who always ruin yourself through being too late, are for once too early." As he stands Tristram has the sudden realization that the old man is "his own figure", but aged, and whose eyes seem to reveal a "bitter personal antagonism". He feels an urge to strike the old man, who raises his hands as if to shield his head, and says "We shall meet later". This is the last thing he remembers before waking up on the floor of the hall of the hotel, the concierge standing over him and saying "You were asking me for someone, sir, when you fainted. Shall I call a doctor?" (p. 220-221)
The next day he leaves Granada, and within a few days is back in London, where he tells the story to the narrator. Then he sees a "nerve-specialist" who reassures him the experience was a dream that happened after he fainted, and so he stops worrying about it. Soon after the narrator realizes that a change has come over Tristram—he basically becomes a sellout, dropping all his old artistic friends, and begins to churn out lower-quality writing that has great popular appeal, making him famous and wealthy (and also receiving a knighthood). The novel jumps ahead to the future period when the narrator is an old man and writing all these recollections, earlier established as "some forty odd years" after the main events of the story in the 1920s (p. 98), so set well in the future of when Sitwell was actually writing the novel (the future setting doesn't play much of a role though there is a brief reference on p. 233 to Tristram's writings being used for propaganda purposes during 'the World War of 1953-1957').
Tristram unexpectedly asks to meet with the narrator again, and seems to show some regret at the course his life his taken, saying that he wants to write a new book which will be his "best work" and re-establish himself as a serious writer. (p. 235) But later it's shown that he is suffering from writer's block, and sleeping badly again, so his doctors order "Rest, change of air, and change of scene" and he goes on another trip through Europe. (p. 260)
He ends up back in Granada, staying (of course) at the Hotel Boabdil. The new concierge seems familiar, but he can't place him. One night he is in his room re-reading one of his earliest novels, when he hears a knock, and the concierge appears, saying "The gentleman to see you, sir." He suddenly realizes that the concierge is a grown-up version of the "whining, piteous child" who always accompanied the Frenchwoman he always saw begging on his earlier trip to Granada. Then he sees the visitor, who seems to be a dead ringer for his own younger self, and the shock apparently kills him. (p. 281-282) None of the dialogue that was spoken in the earlier encounter (or dream) recurs here.
The narrator then discusses the inquest that followed Sir Tristram Orlander's death, noting that multiple witnesses saw a young stranger being brought up to his room, and that the concierge swore under oath that when he asked for a name, the young man had said "Same name—Tristram Orlander—Mr. Tristram Orlander, though. He's expecting me, I believe." The narrator reports this gave rise to the rumor that he had been murdered by an illegitimate son, but the autopsy showed his death hadn't been caused by violence. But he also notes people thought it was odd that the stranger never came forward to "prove his innocence", and no one was able to locate him. (p. 281-282) He also adds that "as, day by day, I followed the evidence, I was even more struck by the likenesses than by the discrepancies between this and the earlier encounter there, which he had related to me (indeed the very existence of such discrepancies as there were, was, according to his theory, an essential part of any such occurrence)" (p. 282).
So, was this a time travel story? I think there are different possible interpretations, and Sitwell probably meant it to be ambiguous. It could be seen as a time slip story where Tristram physically traveled to the future, or one where he did faint but projected something like an astral body into the future (as I mentioned in this answer, astral projection was a popular plot device in weird tales from the late 19th and early 20th century). The discrepancies between the accounts of events from the younger and older Tristram's perspective could be chalked up to unreliable memories of him and other eyewitnesses, although the narrator noted that this seemed to support some ideas about earlier desires/expectations and later events being distorted echoes of one another, suggesting it was more than just a matter of people misremembering what happened. Maybe the young man who entered the old Tristram's room at the end wasn't his younger self at all, but an illegitimate son as the rumors had it, but the fact that something so much like the earlier vision took place supported the idea discussed earlier about drawing events from the future like a magnet if you "wanted or dreaded something sufficiently". In each of these interpretations there could at least be said to be something like a causal loop where the younger self and the older self were each influencing each other's sense of meeting the other, whether or not the younger self had really traveled (in body or spirit) to the future.
