Southampton dismissed Pellegrino yesterday after just one victory in 17 league games under the manager.
That latest setback left Saints languishing just one place and one point above the relegation zone.
Hughes has taken charge of almost 450 Premier League matches for Blackburn, Manchester City, Fulham, QPR and Stoke.More news: US Defence Secretary arrives in Afghanistan
The team are one point above the relegation zone after winning only once in 17 Premier League games and there is a strong feeling that they now need a manager who will have an impact, but potentially also fit with a longer term playing philosophy that gives younger players their chance.
Pellegrino paid the price for overseeing a disastrous run of results that left Southampton in huge relegation danger and was given the boot by the south coast club.
"He has never been relegated as a Premier League manager".
Ex-Watford boss Marco Silva and former West Ham United head coach Slaven Bilic are the other frontrunners in the betting to take over at St Mary's Stadium.More news: Nicole Kidman sets new series 'The Undoing' at HBO
"The club will look to appoint a new management team as soon as possible, with the search for a replacement already underway".
"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 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.More news: Defence minister Nirmala Sitharaman to visit China next month
Southampton are hoping to make a quick appointment in time for their FA Cup clash with Wigan on Sunday.