The 26-year-old forward joined Southampton 2019 from Birmingham City and has made 145 appearances for the Saints.
Adams was a regular under former bosses Ralph Hasenhuttl and Nathan Jones, but since Ruben Selles took charge, he has seen his game time limited.
Southampton’s relegation to the Championship has sparked interest in the forward and Premier League outfits are in the chase for his signature.
Everton are long time admirers of the Scottish international and they explored a move for Adams in the January transfer window.
Nottingham Forest have now joined the race and it has been claimed that they are keen on Adams.
Adams is set to enter the final year of his contract with Southampton and Steve Cooper, who is an admirer of the 26-year-old, wants to bring the player to the City Ground in the summer.
Southampton might agree to sell the forward this summer as they risk losing the player on a free transfer.
And it remains to be seen whether Nottingham Forest will face competition from Everton in their attempt to lure the forward to the City Ground.