To the best of my knowledge, there are no laws requiring a reporter to abide by the rules of "off the record". There is a widely agreed code of professional ethics which says that the reporter should not disclose a confidential source.
If a reporter breaks this code, I do not think there is any way for the source to have a valid tort claim against the reporter, or to sue with a reasonable chance of success on any theory.
But that reporter may find it much harder to get insiders to provide confidential information in future. The reporter may lose respect from other journalists. If the reporter's boss strongly values the professional code, the reporter might get fired. Other media outlets might be less willing to hire the reporter in future. But none of these are required by the law. They are just the possible results of people's personal reactions.
The answer by user6726 suggests that a promise not to reveal a source might be enforceable via contract law. It might be, although I am not aware of a case other than the one cited in that answer, where this has been tried, and I am not sure if the consideration would be sufficient to make an enforceable contrast. But I would not want this answer read as contradicting the other one.
In Cohen v. Cowles Media Co., 501 U.S. 663 (1991) (the case cited in the other answer) a person who provided information to a newspaper on a promise of confidentiality sued when his name was published. A jury awarded $200,000 compensatory damages, and $500,000 punitive damages. The Minnesota State Court of Appeals threw out the punitive award, because fraud was not established, Then the Minnesota State Supreme Court ruled that "a contract cause of action is inappropriate for these particular circumstances." That court then considered that a different theory of the case, promissory estoppel, might allow the judgement to stand. However, the state Supreme Court believed that the Federal First Amendment would bar such an award. That issue was taken to the US Supreme Court, which held that such an award was not barred by the First Amendment. It did not, however, reinstate the $200,000 award, nor rule on the promissory estoppel claim, as that is a matter of state law. Instead it sent the matter back to the state court for further proceedings. The US Supreme Court case record does not show what the final outcome was, but the case does not seem to support the contention that a contract theory can apply to such a situation, at least in that state. But it does suggest that a promissory estoppel case might work, at least in Minnesota. I have not searched for the final outcome in state court.
By the way, it seems from the case record that the information provided by the source, a political campaign employee, was misleading, although that does not seem to have been raised as a defense.