Not a standard feature, no. Most of them don’t support TRRS jacks, but if you look real hard a few models like the FiiO BTR3 at least give you a mic of their own, and the only challenge then will be mounting it somehow close to your mouth so you’re heard as well as with a headset’s boom mic.
I tried this at work for the wireless convenience about a decade back, but the quality I was sending out was… tragic. Bluetooth’s Headset Profile (HSP) transmits only mono and nothing near the quality of the codecs for stereo music (with some recent improvements).