The Athletic says United have also held talks with Atletico Madrid about Morata.
United walked away from possibly signing Mark Arnautovic in midweek, partly due to the backlash from fans.
And now signing Morata is a "genuine possibility".
Talks have taken place between the Red Devils and Atletico over a deal for the 29-year-old Spaniard.
The LaLiga side would be willing to sell for around £30million.
A loan switch is also a possibility, but former Chelsea striker Morata is expected to demand a large weekly wage if he is recruited by United.