Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$? 


You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A\_\infty$-spaces such that the induced monoid on $\pi\_0$ is a group (this has been discussed before, see for example [this question](https://mathoverflow.net/questions/41525/how-should-i-think-about-delooping).) 



Criterion for deloopable based map
As Mark Grant says, $f$ has to be an $A\_\infty$-map. Depending on your situation, it may also be useful to look at it in the following way:


The classifying space $BG$ of the loop space $G$ has a skeletal filtration $B^{(n)}G$ coming, for instance, from the $n$-skeleton of the simplicial space $NG$, the nerve of $G$. (NB: I do not mean the $n$-skeleton of the CW-complex $BG$.) We have that $B^{(0)}G$ is a point and $B^{(1)}G = \Sigma G$. So you always get a map
$$
\Sigma f\colon B^{(1)}G = \Sigma G → \Sigma H =B^{(1)}H \hookrightarrow BH,
$$
and $f$ deloops if you can lift it from $\Sigma G$ all the way along $B^{(2)}G$ etc. to $BG$. This gives rise to an obstruction theory where, depending on your application, you might be able to explicitly compute the obstruction groups.


