Shaw is thought to be close to agreeing a deal to join Manchester United and has even travelled North for a medical with The Red Devils.
Southampton were keen to keep hold of their promising star but a £34 million offer from United is too much to refuse and The Saints will look to replace the English fullback with Kolodziejczak.
The Nice defender - often referred to as Kolo played 33 times last season scoring twice in a successful campaign that has seen him become a target of two Premier League sides.
West Ham United are believed to have already had a €3 million offer for the left-back rejected but remain keen, though Southampton are hoping to beat The Hammers to his signature following the impending departure of Shaw.
A firm offer in the region of £6 million could be enough to tempt Nice into selling the former-Lyon man, who still has two years remaining on his current contract with the Ligue 1 club.
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Kolo can also play at centre-back and he didn't perform too differently to Luke Shaw last season, playing just one minute more than The Saints defender.
He score two goals, but created 22 less chances than Shaw for his teammates, though he was more accurate with his passing.
The Frenchman, who has represented Les Bleus at all youth levels up to the U20's, also won more tackles per game on average and has to be seen as a solid replacement if Shaw's departure is confirmed.
Southampton fans will undoubtedly be disappointed if their promising young star departs as expected, but the arrival of another young defender, with plenty of the Shaw money to spend as well would make it a bit easier to accept.