Gallagher has spent the season on loan at Blackburn Rovers, scoring 12 goals in 47 appearances, but he is now due back at St Mary's ahead of the next campaign.
However, with just one year remaining on his existing deal and the Saints well stocked for forwards, he could be made available for transfer and it appears that he would have a number of admirers.
According to The Sun, Cardiff have identified the 21-year-old as one of their major signings, with manager Neil Warnock keen to mount a promotion charge in the Championship.
The former England Under-20 international is likely to have other offers, but it has been suggested that the Bluebirds currently lead the race for his signature.
Gallagher has made 20 first-team appearances for Southampton, but just three of those have come from the starting lineup in the Premier League.