Le Tissier added the Virgil van Dijk saga in the summer, where the club refused to sell the defender to Liverpool before a big-money move to Anfield in January, may have disrupted Southampton in the first half of the season.
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.
The 46-year-old has paid the price for a dismal run of form that has seen the Saints win just one of their past 17 matches.
"We would like to place on record our thanks to Mauricio, Carlos and Xavier for their efforts during their time with Southampton, and wish them well for the future".
Southampton next manager odds: The favourites to replace sacked boss Mauricio Pellegrino
"The club will look to appoint a new management team as soon as possible, with the search for a replacement already underway".
Despite having being sacked by the FA in September 2017, the former Bolton boss was back in work in December 2017 taking over at Crystal Palace, where he ultimately helped achieve a 14th place finish for the Eagles.
Silva impressed many with his ultimately fruitless attempts to guide Hull City to Premier League safety a year ago and he made a good start at Watford this season until an approach from Everton soured his relations with the club.
"Yes, you can lay the blame at the manager's door at how the team has played and been set but, but at the end of the day the players out on the football pitch have got to do the business".