The Netherlands international has now been reintegrated into the first-team picture at St Mary's having been training with the development squad after handing in a transfer request at the start of August.
Van Dijk had made it clear he wanted the club to take up the interest from Liverpool, who issued a public apology in June following suggestions of an improper approach for the centre-back.
Van Dijk played in the Premier League matches against Crystal Palace and Stoke before the international break, now appearing to be back in Mauricio Pellegrino's plans.
Despite the apparent reconciliation, the 26-year-old revealed during this international break that he has not given up hope of a move in January.
Reigning Premier League champions Chelsea are also believed to be in the running to sign Van Dijk but Le Saux, who spent two years with the Saints after joining from the Londoners in 2003, feels it would be a mistake for him to go to Stamford Bridge.
He said: "If Chelsea are to carry on with playing a back three then they need top quality players in those positions but [Gary] Cahill, [David] Luiz and [Cesar] Azpilicueta are all having decent seasons so far.
"In addition, [Andreas] Christensen has done well having come through the ranks so I think Van Dijk would have quite the fight on his hands in trying to displace those players from the starting XI.
"The bigger issue for me is the need for a strong back-up option to [left- wing-back] Marcos Alonso."