The 21-year-old midfielder is currently owned by Benfica but is on loan at Valencia, where he has been building on his burgeoning reputation.
The newly rich Spaniards are believed to be pushing to make the deal a permanent one in the not too distant future, but the youngster has not yet put pen to paper.
Gomes himself is thought to be aware that his stock is rising, and could hold off in a bid to establish whether or not interest in his signature could lead to a lucrative move.
One issue could be third party ownership, but both Chelsea and United have successfully negotiated deals for players in such situations in recent seasons.
Gomes impressed last season with Benfica, helping the Lisbon giants to claim the Portuguese title and reach the Europa League Final.
He has also won his first senior international call-up and two caps recently.