The Sun says the 19-year-old attacking midfielder is out-of-contract this summer, and free to speak to clubs across Europe this month.
The England youth international’s reps have held talks with sides in Spain and Italy, while Chelsea and Leicester have shown interest.
United are desperate to keep hold of one of their prize assets, and are keen for him to pen fresh terms within the next few weeks.
It is understood Gomes is on around £10,000-a-week at present, and the Red Devils would be willing to boost that to around £30,000-a-week.
Old Trafford chiefs would also be open to allowing him to leave on loan, to gain valuable first-team experience next season.