Hughes, who was sacked by Stoke six days into the New Year, made 52 Premier League appearances for Southampton between 1998 and 2000 under Dave Jones then Glenn Hoddle, scoring just two goals before moving on to Everton.
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.
It is added that former Hull and Watford boss Silva "is not understood to be in consideration" for the post, while ex-West Ham manager Slaven Bilic "is not prominent on the immediate shortlist".
As The Guardian wrote at the time, Puel "did not fall short of the reasonable expectations that Southampton could have had when they appointed him less than a year ago, he just failed to meet their admirably unreasonable ones".
Southampton, who sold centre-back Virgil van Dijk for a record-breaking £75 million ($102 million, 84 million euros) in January, have started leaking goals in recent weeks, conceding 19 in their last 10 Premier League outings. As for Puel, he is now in charge at Leicester who, ironically, are eighth in the Premier League, the position that Saints were when they fired the Frenchman.
"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".
Saints hope to have a new manager in place before facing Wigan on Sunday for a place in the FA Cup semi-finals.
Southampton should appoint Mark Hughes as their new manager until the end of the season, says Robbie Savage.