Saturday's 3-0 defeat by Newcastle United had left them a place and a point above the relegation zone and sealed the Argentine's fate.
Le Tissier suggested that the turmoil surrounding Dutchman Van Dijk's future amid interest from Liverpool and Manchester City had affected the team's performances early in the season.
Van Dijk eventually moved to Liverpool in January for 75 million pounds ($104.24 million), according to British media reports.
"I think that was probably something that, looking back with hindsight, the club would probably regret, trying to keep him in the summer," said Le Tissier, who scored 209 goals for Southampton during his 16 years at the club.
"They tried to do the right thing by the fans, in terms of not selling their best player every season, but sadly that did appear to backfire, and that situation probably didn't help the performances in the first half of the season."
Southampton have lost a large number of players to their domestic rivals in the past few years and club chairman Ralph Krueger said earlier in the season that they were keen to end the perception that they were a selling club.
"That's life, that's football, those decisions were made with the best intentions and it didn't turn out right," Le Tissier said. "It's now time to move on, appoint a new manager, and get out of the trouble we are in at the moment."
Welshman Mark Hughes was firming as the bookmakers' favourite to replace Mauricio Pellegrino as the south coast club's manager.
($1 = 0.7195 pounds)
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
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