Manchester United sold Keane to Burnley in 2015 for £2.5million after being told by Louis van Gaal he was surplus to requirements.
Since then, Keane has seen his stock rise at Turf Moor.
Jose Mourinho is set to embark on a big spending spree and is intent on strengthening his defence.
How Man United could look after £250m summer spending spree Thu, April 13, 2017
Is this how Manchester United could look if Jose Mourinho goes on a £250million summer spending spree?
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How Manchester United could look if Mourinho splashes out this summer
Michael Keane has no shortage of admirers
Jose Mourinho is keen to bring Michael Keane back to Old Trafford
Chelsea and Liverpool have also emerged as suitors, with Antonio Conte and Jurgen Klopp respective admirers of the 24-year-old.
Initially, it was thought Keane would cost £25m.
United are believed to have a sell-on clause in Keane’s contract.
But the Daily Mirror claim Keane would now cost United £30m should they wish to re-sign him.
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Leicester tried to sign Keane last summer, but failed with a £20m bid.