The 22-year-old has enjoyed a fine campaign at St Andrews this season, and had attracted interest from the top flight over a potential move last month.
However, according to The Times, the Blues are seeking around £22m for Adams signature, and rejected a £12m approach from Burnley despite the clubs financial concerns.
It was also earlier reported that Burnleys Premier League rivals West Ham and Southampton had registered their interest in making a move for the forward, but were turned down by Birmingham.
The 22-year-old earned a high-profile move to Sheffield United in League One in 2014 after catching the eye of scouts during his playing days at non-league level.
After scoring just three times in his first season at Bramall Lane, Adams fared better in his second season at the club, finding the net 12 times in 41 appearances.
The England U20 internationals form saw him join Birmingham in a reported £2m deal, and scored a combined 16 goals in his first two campaigns at St Andrews.
This season however, Adams has enjoyed his most prolific season in professional football to date, with 19 goals and a further three assists in 31 Championship games, including a first-half hat-trick against QPR on Saturday.