Yuval Filmus shows in [Link](https://doi.org/10.1007/s00236-003-0107-x) that it is decidable for a regular language $L$ whether $w\in L\implies w^k\in L$ for all $k>0$.  I would guess there must be an older reference.  On the other hand, a language $L$ is called pure if  $w^k\in L\implies w\in L$.  It was shown by Pedro Silva that purity is decidable for regular languages in http://cmup.fc.up.pt/cmup/preprints/2002-18.pdf

Since the notion of circular language in the question is the conjunction of these two properties, it is decidable if a regular language is circular.