Sky Sports reported that Southampton are yet to make a formal move for Adams, but they have been closely monitoring the 22-year-old.
The Mirror, however, reported that they have had an £8m approach for the striker rejected by the Championship club.
The Saints are now weighing up an improved offer, while Burnley are expected to table their first offer shortly. It has been reported that Birmingham boss Garry Monk is determined to keep hold of his star player, but the club will have little option but to cash in should they receive a fee in the region of £15 million.
Adams, who joined the club from Sheffield United in 2016, has scored 14 goals in 30 games for Birmingham this season.