Wolves' Bakary Sako has revealed he is mulling over offers from the Premier League but is yet to decide on his future.
Rotherham boss Steve Evans will decide the fate of his out-of-contract players in a meeting on Monday.
Centre-half Gardos, 25, is expected to be a like-for-like replacement for Dejan Lovren after the Croatia international joined former Saints captain Adam Lallana at Liverpool.
New manager Ronald Koeman has seen a host of last season's Southampton stars sold off to Premier League rivals before he began recruiting his own targets to St Mary's.
The Dutchman has completed the signings of left-back Ryan Bertrand on loan from Chelsea, Saphir Taider in a loan swap for club-record signing Dani Osvaldo with Inter Milan and England goalkeeper Fraser Forster from Celtic.
Koeman is still trying to complete a deal for Leeds right-back Sam Byram with the West Yorkshire club rejecting an initial offer in the region of £4.5million for their player.
However, Saints could lose out to Manchester United for Sporting Lisbon's Argentina left-back Marcos Rojo after the defender fuelled speculation linking him with a move to Old Trafford over the weekend by following a handful of United players on social media.