Fiorentina president Andrea Della Valle is hopeful that Chelsea target Juan Cuadrado will not leave in January.
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.