Some observers were surprised when Southampton fired manager Claude Puel last summer after he guided the team to an eighth-place finish in the EPL and to the final of the EFL Cup, where they lost 2-3 to Manchester United.
Southampton are hoping to have Pellegrino's successor in place for Sunday's FA Cup quarter-final against Wigan and Press Association Sport understands talks are under way with former Saints player Hughes.
It is believed Southampton aren't set to appoint Hughes imminently, while other candidates such as former Hull City and Watford manager Marco Silva, plus ex-West Ham boss Slaven Bilic, aren't now in the running.
Mark Hughes, who had a short spell at Southampton as a player, is one of the leading favourites to replace Pellegrino.
The club sit 17th in the league on 28 points, a point ahead of Crystal Palace and Stoke, with eight games left to play.
"As you know, my period in Southampton has finished".
"Some moments we managed this, some moments we couldn't, but we have been living a hard season for many reasons".
It was widely believed that Silva would top Southampton's list of replacements for Mauricio Pellegrino, who was sacked by the club on Monday.
As reported by The Scotsman, Celtic manager Brendan Rodgers' achievements in his first two seasons in Glasgow have lead to some strong interest from Premier League club Southampton.