Pellegrino was only appointed as manager last June, but failed to get any sort of consistency going, and Southampton are now in 17th place in the league, having won only 1 of their last 17 matches.
Hughes has been lined up to replace Mauricio Pellegrino, who was sacked by Southampton on Monday.
Southampton are hoping to announce their new manager before they visit third-tier Wigan Athletic in the FA Cup quarter-finals on Sunday with their next league game not until March 31 when they travel to West Ham.
Pellegrino had won only once in 17 league games and his chopping and changing of the team did not lead to him finding the right blend.
Hughes is understood to be keen to return to management and believes Southampton to be a well-run club blessed with talented players that he had targeted - and missed out on - during his time with Stoke.
Hughes, out of work after being sacked by Stoke earlier this year, has good links with the club having played for them for two seasons up to 2000.More news: One of Lloyd Blankfein's heir apparents just unexpectedly retired
More news: South Park: The Fractured But Whole is coming to the Switch
More news: Percent Increase in Roseville Gas Prices Since 2016
"As you know, my period in Southampton has finished".
"For this reason, I would like to say sorry to our fans because we couldn't represent our club properly".
Argentine coach, Pellegrino was appointed 9 months ago, but a 3-0 defeat in the hands of Newcastle on Saturday, seems to have informed the club's decision to end its relationship with the former Argentine global.
Pellegrino said: "The manager is the face of a lot of decisions and actions but I always tried to put the club first with my technical staff and the board behind me".
Of this season's contrasting fortunes, Pellegrino said: "We have been living a hard season for many reasons".
The 54-year-old's extensive experience in the Premier League is attractive to Southampton, whose need for an upturn in results is urgent.