The 33-year-old appears set to leave Staffordshire following seven years there, with the offer of a longer contract in Lancashire something which is appealing to the Republic of Ireland international.
Back in 2015 Stoke had been initially reluctant to offer Walters a two-and-a-half-year deal, but they eventually did, meaning he is under contract until next summer.
But Hughes is now prepared to green light his move to Turf Moor given their more favourable terms, leaving Walters free to join his international compatriots Robbie Brady, Jeff Hendrick, Kevin Long, and Stephen Ward.
“It seems that maybe Jon could take the option of a move to Burnley,” Hughes said.
“I haven’t had the opportunity to speak to him about it. I think it’s something that maybe he wants to make happen, but irrespective of that, if it falls through we would be more than happy because Jon coming back to the club would be fine.
“Maybe at this point of his career Jon feels that he needs a bit more security in terms of length of contract which maybe we weren’t prepared to match what he’s got on the table. If he does go, we will look back on his time here with fond memories.
“He’s been a huge part of the success of the club in the Premier League and his contribution will never be underestimated.”
Meanwhile, Walters’ Ireland team-mate Shane Long intends to make up for lost time and “hit the ground running” under new Southampton manager Mauricio Pellegrino next season.
The 30-year-old missed the closing weeks of the last Premier League campaign with a broken metatarsal bone in his foot. Following the departure of Claude Puel, it is a fresh start for everyone at St Mary’s Stadium.
Long hopes he will be able to use the lead up to the new season, which will start at home to Swansea on August 12, to show Pellegrino just what he can still offer the team.
“For me, last season was a bit disappointing,” Long, who was often used as an impact player from the bench or in the Europa League under French coach Puel, said.
“I didn’t play as much as I would have liked to and felt I didn’t get the chances that I deserved at times, so I want to hit the ground running this year. It’s the first pre-season in a long time that I’ve been here from the start, so I want to show the manager what I have got and leave an impression on him come the start of the season.”