It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-theory which represents an additive bivariant functor on “separable C*-algebras” which is a common generalization both of singular homology and de Rham (or singular) cohomology ?
Yes. There is the "singular chains" functor $C_*:\mathcal{S} \to \mathcal{D}(\mathbb{Z})$, where $\mathcal{S}$ denotes your favorite category of spaces, and $\mathcal{D}(\mathbb{Z})$ is the derived category. $H_*(X)$ is just computed as homology groups of $C_*(X)$, while $H^*(X)$ is computed as homology groups of $\mathrm{RHom}(C_*(X),\mathbb{Z})$. So the "bivariant homology" you're looking for is given by (the homology groups of) $\mathrm{RHom}(C_*(X),C_*(Y))$.
Let me give some references for bivariant "de Rham theory" for locally convex and $C^*$-algebras. We know that, roughly speaking, Hochschild homology (equipped with an $S^1$-action) is a "non-commutative" analogue of de Rham complex, and consequently, periodic cyclic homology is a "non-commutative" analogue of de Rham cohomology. There is a bivariant version of Hochschild and cyclic homology, introduced in
Bivariant cyclic theory, John D. S. Jones, Christian Kassel, K-Theory v. 3, n. 4 (1989): 339-365
The original work concerns the algebraic theory, if I understand correctly, but later works extend this to locally convex and $C^*$-algebras. There are some surveys. For example, the following survey explains the (local) cyclic theory for locally convex algebras and $C^*$-algebras:
Cuntz, J. (2004). Cyclic Theory and the Bivariant Chern-Connes Character. In: Doplicher, S., Longo, R. (eds) Noncommutative Geometry. Lecture Notes in Mathematics, vol 1831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39702-1_2
