The 22-year-old, who is currently preparing for the World Cup with England, has emerged as a potential target for Jose Mourinho's side, according to reports.
The Portuguese coach is reportedly looking at defensive options in the transfer window, with David Luiz being linked with a move to Barcelona and John Terry moving towards the end of his career, despite recently signing a one-year extension.
Jones did not enjoy a successful campaign under David Moyes at Old Trafford, but it is understood that the former Blackburn Rovers man would provide cover in a variety of positions if a move to Stamford Bridge does materialise.
United are unlikely to welcome bids for Jones, but it remains to be seen where the youngster fits into the plans should Louis van Gaal take over as expected.
The club would be unlikely to welcome advances for one of their players, though it remains to be seen where Jones fits into incoming manager Louis van Gaal's plans.
Jones joined United in 2011 for £16.5m and has made 104 appearances for the club.