Stekelenberg, who last plied his trade at Ligue 1 club Monaco, will be available on a free transfer.
The former Roma 'keeper has earned 54 caps for the Netherlands during his career and looks to be a handy replacement for the crocked Fraser Forster.
Forster injured himself against Burnley earlier in March after having an impressive start to his tenure in between the Saints sticks.
Meanwhile, Southampton let Artur Boruc leave the club a week ago so are in need in goalkeeping reinforcements.
Aging club captain Kelvin Davis has had mixed fortunes since covering for Forster's absence, while young goalkeeper Paulo Gazzaniga has not done anything to prove he is ready for first team action in the Premier League.
Saints boss Ronald Koeman has been successful in delving into a bank of players involved with Dutch football so here's hoping Stekelenburg can keep that trend going.
Rumour rating: 7/10 - I think it's a realistic option and he is more than capable of doing a job. Though the source isn't the most credible.