However, the striker's agent suggests his client will change his mind and consider a move elsewhere if he is not given first-team assurances by Zinedine Zidane ahead of the 2017-18 season. A final decision is expected in the coming days.
Wide-ranging reports suggest United have already seen an opening offer of £52m (€59.9m) rejected by Real Madrid for the 24-year-old. He became United's primary attacking target after they backed away from negotiations for Atletico Madrid's Antoine Griezmann.
Real Madrid are demanding an astonishing £78m for the Spain international, but United are reportedly confident of persuading Real president Florentino Perez to lower that fee.
Morata was limited to just 14 starting appearances as Real sealed a 33rd La Liga crown last term and despite his excellent return of 15 goals in the league, it remains to be seen whether Zidane sees him fit to displace Karim Benzema.
The former Juventus forward admits securing game time is a must ahead of the 2018 World Cup next summer, but says his priority is to get those opportunities in the Spanish capital.
"I want to be playing and to be important to the squad, particularly next year when there is the World Cup," Morata said, the Mail report. "But I want to fight for my place here, I would be crazy to leave Real Madrid."
But speaking to Foot Mercato, Morata's agent insists that above all else, his client's priority is to play – be it at Real or elsewhere, with a decision expected in coming days.
"He does not want to do a season like this again. It was a great season, but he wants to play more. Alvaro is a player who wants to play a more important role. He wants to play more. Which is normal for an attacker who has scored 19 goals. It is too little.
He added: "The player will take a clear and definitive decision in the coming days. I have not yet met with representatives from Real Madrid."
United have made bringing in a more traditional number nine a top priority this summer, putting their interest in Griezmann on ice with Atletico Madrid's transfer ban also contributing to a deal being shelved for at least a year.
United have also been linked with Romelu Lukaku, Andrea Belotti and Kylian Mbappe as they seek to bring in a long-term replacement for Zlatan Ibrahimovic, who could still leave the club this summer when his contract expires.