United have been chasing Morata, 24, since Antoine Griezmann turned down a move to Old Trafford earlier this summer.
He is set to leave the Bernabeu after struggling for game time under Zinedine Zidane last season.
The Spanish forward made 26 appearances in the league but just 14 of those came as a starter.
Morata was also only used as a late substitute as Real ran out 4-1 winners over Juventus in the Champions League final.
Bernabeu bosses rejected United's opening bid and were holding out for a fee in the region of £75m-£80m.
But, as Starsport revealed last week, Morata took time away from his honeymoon to try and push through a deal.
And Marca believe he is now on the verge of getting his wish.
The Spanish newspaper, which has close ties to Real, say the latest bid from United is worth around £66m.
And they reckon that may well do the trick, claiming a deal could be reached in "the next few hours".
Jose Mourinho is a long-time admirer of Morata, having handed him his Real debut during his time in charge at the Bernabeu seven years ago.
And he is said to have maintained regular contact with the former Juventus star in recent weeks to reassure him that his move to Old Trafford will eventually be completed.
