The Reds captain, out of contract next month, is mulling over his future, with the Potters viewing the 26-year-old as a potential long-term successor to Jack Butland, tipped to return to the Premier League this summer.
Birmingham City, Hull City, Bristol City and Brentford have also been credited with an interest in Davies, alongside top-flight outfit Wolves, in the market for a back-up goalkeeper.
Davies – named in the PFA League One team of the year side recently – will hold discussions with the Reds, who still harbour hopes of retaining his services, shortly.
Speaking about his future last week, the Wales international said: “We will have a sit down with the board and stuff and see where we are with them.
“I am open-minded. I will take some time and have a good think and then we will see where we go.”
Meanwhile, attacking midfielder Mike-Steven Bahre is confident that he will remain at Oakwell next season, with talks having been held to keep him at the club.
The Hannover 96 player has enjoyed an impressive season-long loan at the club and is keen to test himself in the Championship in 2019-20.
He said: “I would like to stay. There has been nothing written yet and no contract, but I think we are working on that and I think I will stay next season in the Championship.”
The 23-year-old admits that being part of Barnsley’s memorable campaign has represented the best moment of his career so far, with the feat eclipsing his experiences with his parent club when they were promoted to the Bundesliga in 2017.
Speaking after Saturday’s game at Bristol Rovers, he added: “It could not be better. When I came here nine months ago, this was our big target and now we have achieved it, it is unbelievable.
“It is one of the best days of my life and best of my career.
“I was promoted two years ago, but I did not feel a part of it as I was injured all season.
“Here, I feel I have been a big part of the promotion. So you cannot compare.
“My family and friends have been over here watching games and I have seen many German flags over here.”