Southampton is in 17th place, a point and a place above the relegation zone.
The ex-Watford boss was immediately made the stand-out favourite to take over at St.Mary's stadium after Pellegrino was relieved of his managerial duties on Monday, but reports have since suggested otherwise.
Saturday's 3-0 defeat at Newcastle left the Saints just a point outside the relegation places with only eight games of the Premier League season remaining.
Two of Pellegrino's top assistants, Carlos Compagnucci and Xavier Tamarit, were also let go.
"There are all sorts of managers mentioned; Mark Hughes has a lot of Premier League experience and I think that will count in his favour".
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"The club will look to appoint a new management team as soon as possible, with the search for a replacement already underway", Southampton said.
Hughes has been out of work since being sacked by Stoke City in January, but the club had previously recorded three consecutive finishes of ninth before sliding last season to 13.
"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.
Southampton are hoping to announce their new manager before they visit third-tier Wigan Athletic in the FA Cup quarter-finals on Sunday with their next league game not until 31 March when they travel to West Ham.
Le Tissier, who played 443 league games for Southampton over a 17-year stint, added blame should not just lie at the door of the manager, and wants the next boss to bring fresh impetus to the under-performing players.