Morata, 24, is set to leave Real Madrid after struggling for game time at the Bernaebu last season.
The forward made 26 appearances in the league but just 14 of those came as a starter.
He was also only used as a late substitute as Zinedine Zidane's side ran out 4-1 winners over Juventus in the Champions League final.
Jose Mourinho is a long-time admirer of Morata, having handed him his Real debut during his time in charge of the Spanish giants in 2010.
But Bernabeu bosses rejected United's opening bid and were holding out for a fee in the region of £75m-£80m.
However, as Starsport revealed last week, Morata took time away from his honeymoon to try and push through an Old Trafford switch.
And The Sun say significant progress was made over the weekend after Morata's father Alfonso and agent Juanma Lopez held positive talks with the club.
They claim Real have now backed down on their asking price for the Spain international and are willing to do business at United's £60m valuation.
The Sun believe the Red Devils are hopeful of officially announcing the deal on Thursday.
Morata will become their second signing of the summer after Victor Lindelof's £31m arrival from Benfica.
