I dealt with this yesterday for a staff member. He was using a Mac and was getting the prompt for a keychain password. Once we resolved that he then tried logging into Office. He not only kept getting prompted but also was getting an error from the provided numbers to put into the MS Authenticator saying they were invalid. Double checked the time on both devices, Company portal was synced, all apps were up to date and no policies have changed that would have caused that.
Lone behold we switched to Edge and it was working no problem. At this time we believe the culprit to be a Chrome setting that doesn't like MS Authenticator for whatever reason.