With Saints languishing 17th in the Premier League having managed just one win in 17 top-flight matches, the club sacked former Argentina international Mauricio Pellegrino on Monday evening.
Southampton are hoping to have Pellegrino's successor in place for Sunday's FA Cup quarter-final against Wigan and talks are under way with former Saints player Hughes, who was backed for the post soon after the market opened.
BBC Sport are now reporting that Hughes could be named as new boss on Wednesday.
The Welshman was sacked by Stoke in January after four-and-a-half years, having previously managed Wales, Blackburn, Manchester City, Fulham and QPR.
Negotiations on a deal that suits Saints and Hughes are understood to have started and Sky Bet make him 1/7 to take the job.
Meanwhile, outgoing manager Pellegrino has sent an open letter to fans.
"I feel so sorry to leave the club in this situation," the Argentinian wrote on the club's official website.
"But when you don't get results I understand the decision the club has to take.
"As you know, my period in Southampton has finished. From the beginning I tried to do my best every single day, I tried to give back to the club and tried to represent the club with pride.
"Some moments we managed this, some moments we couldn't, but we have been living a difficult season for many reasons.
"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.
"The fans have been behind us even in difficult games and I am really grateful for your support in those situations.
"I have to say thanks to Ralph (Krueger, club chairman), to the board, to Les (Reed, executive director) and Ross (Wilson, director of recruitment and scouting), all my technical staff and all the Southampton employees. Also to my players, who respected me in every single moment.
"I wish everyone all the best for the rest of the season, I have enjoyed an amazing experience and I know that Southampton has everything required to move forward."
Southampton were thrashed 3-0 by fellow strugglers Newcastle at the weekend and head to embattled West Ham on March 31 in their next Premier League match.
Hughes played for Saints between 1998 and 2000, making 54 appearances in the Premier League.
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