The Daily Telegraph's Matt Law reported as much on Friday after the Spaniard "agreed in principle to the terms that will be offered to him."
The Red Devils are set to go back with a £60 million bid for the forward after failing with an offer of £52.3 million.
Morata appears to be confident of the move taking place, with Law adding he "has told friends that he expects" to join the UEFA Europa League holders this summer.
Per Marca's Jose Felix Diaz (adapted by Euan McTear), Morata's agent Juanma Lopez told Real director general Jose Angel Sanchez on Friday his player wants to move on this summer.
The 24-year-old is hoping for first-team football next season with the view of getting in Spain's starting lineup at the 2018 FIFA World Cup.
Per Diaz, Lopez said: "He doesn't want to experience another season like that. It was a great season, but he wants to play more."
Morata was restricted to appearances from the bench in Madrid last season, with manager Zinedine Zidane often opting for Karim Benzema despite the Frenchman's poor campaign by his standards.
The younger forward made 43 appearances in all competitions but spent just 1,872 minutes on the pitch, an average of 44 minutes per outing.
Despite that, he notched 20 goals and six assists. Per OptaJoe, Morata is a ruthless finisher and showed as much in La Liga:
27.3% - Álvaro Morata scored 27.3% of his shots in La Liga last season, the 2nd best % of the 59 players to score more than 5 goals. Devil. https://t.co/w91beNGgb82017-6-9 12:34:08
He compares well with United's attacking talent, per Sky Sports News HQ:
Sky Sports News HQ @SkySportsNewsHQ
.@ManUtd confident of landing Alvaro Morata as talks continue with @realmadrid, #SSNHQ understands: https://t.co/ykJWjJp5HU https://t.co/tfLx6M8gmy2017-6-7 19:26:09
Morata has always looked an impressive player, and he was known for scoring goals in big games at Juventus—striking in the 2015 UEFA Champions League final against Barcelona, for example—but he has put to bed any lingering concerns that he isn't a prolific goalscorer, comfortably surpassing his previous best tally of 15 for a season.
Nevertheless, Real's reported valuation of him is excessive, per Sky Sports News HQ's Kaveh Solhekol:
Kaveh Solhekol @SkyKaveh
Real Madrid bought Alvaro Morata back last summer using €30m (£26m) option. Hard to see why they now think he's worth €90m (£78m)2017-6-8 17:10:50
While £60 million would still fall some way short of that enormous fee, it's a generous offer for a player who isn't a regular starter.
The question now is how hard a bargain Real are prepared to drive.