universal bundle of topological monoids

Question

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such that $EG\simeq *$. For general topological monoid, is there such a space $EM$ and a map $$ p: EM\to BM$$ such that $p^{-1}(b)\simeq M$ for each $b\in BM$?