With Fraser Forster recovering from a serious knee injury, Ronald Koeman is said to be interested in the Dutchman.
The Saints lost Fraser Forster towards the back end of the season following a serious knee injury, and he could miss the start of the season as a result.
Meanwhile, Fulham are keen to move Stekelenburg on as they face another season in the Championship. He spent time on loan at Monaco last season and they have decided not to make his move to the club permanent after he only made seven appearances during his time in the principality.
Kelvin Davis was the man in goal towards the end of the season with Paul Gazzaniga also used, which saw the Saints finish their Premier League campaign in seventh place, and securing Europa League football for next season.
The extra games at the start of the season are likely to make the need for cover in the goalkeeping department all the more urgent.
What the Saints would get is a goalkeeper who has been at a couple of Europe's big clubs having appeared for both Ajax and Roma - though some will point to his time when he was part of a Fulham side that leaked a lot of goals during their relegation from the Premier League in 2014.