The Spain international has been heavily linked with a move to Old Trafford since United cooled their interest in Atletico Madrid forward Antoine Griezmann.
Morata would appear to be considering his options after his representative, Juanma Lopez, revealed last week that the 24-year-old could not endure another season like the last one, where he started just 14 games for the newly-crowned La Liga champions.
The Red Devils reportedly had a €60 million offer knocked back by Los Blancos, who are now deciding whether to accept their latest bid, according to Lopez.
“Manchester United? They are a club with great glamour, a very interesting option,” he told Calcio Mercato.
“I can say that their proposal is a very, very important one and now the decision rests with Real. I cannot say [if a deal will be closed soon] – we will see.”
Milan had also made a move for the former Juventus player, but their proposition was well below Real’s expectations.
“Yes, there was frequent contact with us until a few days ago – I can confirm that,” Lopez added. “Quite simply, Real Madrid considered the Rossoneri’s offer to be unsatisfactory.”