The Mirror suggests that Leicester and eleven other Premier League clubs had scouts watching Adams against Middlesbrough at the weekend.
Adams has 13 goals this season, his best scoring run.
The likes of Burnley, Wolves, Fulham and Huddersfield want to sign the player.
Leicestershire Live believes City would like to sign Adams, who once had a trial at the club before slipping away.
But it would be tough to prize him from Birmingham without triggering a bidding war with other teams.