Recent speculation has suggested that the two-time England international has decided to leave Old Trafford in January in a bid to gain more experience.
That is said to have alerted a number of clubs, including Newcastle United, West Ham United and his former team Crystal Palace.
However, those outfits will now face competition for the 20-year-old's signature from the Old Lady, who are keen to bolster their attacking options, according to the Daily Star.
Zaha joined United in January, but remained with Palace until the summer on a loan basis. He is yet to feature in a competitive fixture for the Red Devils this season.