The attacking midfielder has suffered a long-term decline in form at Old Trafford and has gone almost 14 months since scoring a goal or registering an assist in the Premier League - with his last coming in December 2018.
The 27-year-old has 18 months remaining on his contract at the Red Devils while they also have an option to extend his deal by a further year.
The England international has appointed super agent Mino Raiola as his new representative which has played a part in these fresh links, although United are said to have held informal talks over a new deal.
The player is said to have off-field personal issues to deal with alongside his struggles for form on the pitch, and United boss Ole Gunnar Solskjaer is also said to be understanding of this situation.