The highly-rated Spaniard has been on Arsenal’s radar for some time, and is rumoured to be in line for a move away from the Bernabeu once the season ends.
United have been alerted by this, and are understood to be closely monitoring the 21-year-old.
Morata has long been hailed as one of Real’s brightest talents, but the player himself is said to be growing frustrated at a lack of first-team opportunities.
With Karim Benzema holding down a starting berth, the front-man is battling with another youngster in the shape of Jese Rodriguez for action – with his rival in excellent form.
United are believed to be considering selling Javier Hernandez in the summer, and could look to replace him with Morata.
Arsenal are thought to have been keen on a loan deal for the striker in January, but their path was blocked by Real’s desire to keep their man.
As well as the Gunners and the Red Devils, both Liverpool and Tottenham have been linked with Morata, who is valued at around £20m.