Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map of dg-vectors spaces such that $gf-id_A\sim 0$ and $fg-id_B\sim 0$.

My question is: could we extend $g$ to an $A_{\infty}$-map $B\to A$? Is it a consequence of the Homotopy Transfer Theorem?