Which homotopy algebras come from transfer?

We characterize $A_{\mathrm{\infty }}$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_{\mathrm{\infty }}$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal{P}}_{\mathrm{\infty }}$-structures over a field of characteristic zero, for any quadratic Koszul operad $\mathcal{P}$.