Adkins was axed last week despite a run of two defeats in 12 games to drag the club out of the Barclays Premier League drop zone, with Mauricio Pochettino immediately installed as his successor.
Pochettino will take charge of Saints for the first time in Monday's home clash with Everton, at which protests are expected with one fans' spokesman warning Pochettino must work hard to gain respect.
Speaking on BBC Radio Five Live's Sportsweek programme, Le Tissier questioned the timing of Adkins' departure and the motivations of Cortese, who inherited the role of chairman from late owner Marcus Liebherr.
Le Tissier said: "I don't know if Nigel was becoming too popular and the chairman didn't like it - he does appear to have a bit of an ego problem.
"It is a little bit bizarre but some of the decisions that have been made have been a little bit bizarre since the current chairman took over. In a way it wasn't a surprise because I think he's been looking to get rid of him for a while now."
Le Tissier paid tribute to Adkins, who led the club to two successive promotions prior to the current season, and further criticised Cortese for refusing the chance to appear in public to explain his decision.
Le Tissier added: "We have lost just two in 12 in the Premier League and looked like a team that had turned the corner and now everything is turned upside-down again. The fans are very appreciative of what Nigel has done. He is very, very well regarded among the players and supporters and sadly for him there was only one man who thought he wasn't doing a good job.
"The strangest thing about it all for me was.. he brings the new manager in about a minute after he sacked Nigel and then he hangs the new manager out to dry at a press conference on his own, which I thought was an incredibly cowardly thing to do.
"If he's going to make those decisions surely he owes it to the fans to sit there and explain his decision but he wasn't brave enough to do that."
- Sports & Recreation