The Saints hope to have their new boss in place before facing Wigan on Sunday for a place in the FA Cup semi-finals.
Whoever is appointed will be the club's fifth permanent manager in the five years since Mauricio Pochettino replaced Nigel Adkins.
One man who knows Hughes well is his former Manchester United teammate turned Sky Sports pundit Gary Neville.
Reacting to the news Pellegrino has been sacked and the fact Hughes had been linked to the role, Neville told Sky Sports it is a gamble by the club.
"They need a hit man to go in there, someone with the experience of being down there in the league, somebody who can get immediate results," Neville said on Sky Sports' Monday Night Football.
"It's not an exact science, we've seen it work at clubs this season, we've seen it fail at clubs this season, and previous seasons.
"It's just a gamble, that's all it is, and with the money at stake, clubs are willing to take that gamble more and more often. They will go for it, and change managers, even at this time."
Saints have also been linked with former West Ham boss Slaven Bilic and ex-Watford manager Marco Silva.
Robbie Savage has backed a move for Hughes while many pundits and fans have highlighted that the South Coast club would have been better served sticking with Claude Puel who has managed an impressive turnaround at Leicester City.