Potters boss Gary Rowett faces competition from Sheffield United and Blackburn Rovers for the striker's signature, according to the Sun.
However, Stoke are at an advantage as they are prepared to buy Gallagher, whereas Sheffield United and Blackburn are only prepared to make loan offers.
Southampton would prefer a sale as they are looking to raise funds in order to launch a bid for Liverpool striker Danny Ings.
While Mark Hughes is set to recruit Ings, he is said to be in no rush to sell Gallagher, having watched him play in Southampton's opening pre-season friendlies.
Gallagher impressed during a loan spell at Birmingham City last season, but news of Ings' potential arrival could mean that Gallagher seeks a move away in order to get regular first team football.
The 22-year-old striker has scored just 21 goals in 116 career appearances to date, and his only piece of silverware to date is the Under-21 Premier League Cup, which he won with Southampton back in the 2014/15 season.
He has had loan spells at MK Dons, Blackburn Rovers and Birmingham City, all of which were not particularly productive, although he did score 12 goals in 47 games for Blackburn.
Stoke boss Rowett was close to signing Gallagher last season when he was in charge of Derby, and could return to the Rams to sign attacking midfielder Tom Lawrence.