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Jones, 29, finally made his return for United in the Red Devils’ 1-0 defeat to Wolves at Old Trafford earlier this month.
Ralf Rangnick partnered Jones alongside Raphael Varane in his injury-hit defence, with Harry Maguire, Eric Bailly and Victor Lindelof all sidelined at the time.
The United defender, who returned from a long-term knee problem, played the full 90 minutes and received a spine-tingling reception from fans after the match.
Prior to the Wolves defeat, Jones had not featured for United since his side’s 6-0 drubbing of Tranmere Rovers in the FA Cup in 2020.
The centre-back has been at United for over a decade and his current deal with the club will expire in 2023.
According to The Daily Star, both Newcastle and Burnley are prepared to make a move for Jones if he decides to exit Old Trafford.
It is believed that the Red Devils are reluctant to sanction the sale of Jones, with United slapping a hefty £15m price tag on the former England international.
Jones could leave United in a bid to secure more regular first-team football after overcoming his injury nightmare.
Legendary former United manager Sir Alex Ferguson brought Jones to Old Trafford from Blackburn Rovers in a deal worth up to £17m in 2011.
Ex-England boss Sam Allardyce worked with Jones while he was managing Blackburn and praised him as “one of the top two young players I have ever worked with.”
Big Sam admitted that the former Blackburn defender’s career at United had not “flourished as much as I would have wanted it to,” but he’s backing Jones to make a “strong comeback.”
The former Bolton and West Ham manager told Footy Accumulators: “His career hasn’t flourished as much as I would have wanted it to at Manchester United.
“But I hope that he’ll make a strong comeback if not at Man United then somewhere else because he deserves to.”