Given an operad $A$ of topological spaces, we consider $A$-monads in a topological category $\mathcal C$. When $A$ is an $A_\infty$-operad, any $A$-monad $K \colon \mathcal C \to \mathcal C$ can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an $A_\infty$-monad and prove that it is an $A_\infty$-monad itself.