Stekelenburg spent last season on loan at St Mary's.
The Daily Echo reports that the 33-year-old is not wanted by the Championship side with Fulham keen to offload the Dutchman, who acted as cover for much of the season as Fraser Forster recovered from an ACL injury.
Kelvin Davis has decided to retire while Paulo Gazzaniga could still be considered as a little too inexperienced. Given that Southampton could have more football to contend with next season after their strong end to the campaign saw them finish in sixth place in the Premier League - with a Europa League campaign to follow next season - there could be more opportunities when a back-up goalkeeper is called upon.
Stekelenburg made a total of 25 appearances in all competitions last season, and given that manager Ronald Koeman has worked with him prior to their link-up at Southampton, when Koeman was in charge of Ajax, it would seem a sensible move to try to retain his services for next season.
A further spell on the south coast could be another loan deal for Stekelenburg, who has not played since Forster's return to the side from his injury.
However, given the trust that Koeman has in him, and that his chances of playing some games are more likely than at Fulham - where it seems he is not well fancied at all - it could be a move to suit all parties ahead of next season.