The centre-back is under contract until the end of the next season, but the Red Devils are yet to trigger a one-year extension on his contract.
The Times claims Allardyce will go after Jones, should he remain at the Toffees helm next season.
Injuries have consistently hampered Jones since he arrived from Blackburn Rovers in 2011. He has made just over 180 appearances in his seven seasons at Old Trafford.
A hamstring injury has disrupted Jones this season, and with Eric Bailly, Chris Smalling, Victor Lindelof and Marcos Rojo available for selection, the 26-year-old has found it hard to solidify a starting spot.
The Englishman hasn't been included in Jose Mourinho's squad for United's previous three Premier League matches.