The Red Devils have been struggling to agree a transfer fee with Real Madrid for the Spain international.
But, according to the IBTimes, they made 'significant progress' over the weekend.
United have already seen an offer in the region of £60m knocked back, with Real holding out for more.
Jose Mourinho wants Morata after cooling his interest in Antoine Griezmann and losing out to Chelsea for Romelu Lukaku.
Starsport reported last week that Morata briefly left his honeymoon in a bid to speed up his move to Old Trafford.
Morata travelled to Madrid to hold talks with Real president Florentino Perez.
The 24-year-old has informed the club he wants to leave after a frustrating campaign in which he started just 19 games in all competitions.
Morata scored 20 goals, despite his limited action, and Zinedine Zidane attempted to persuade him to stay.
But Mourinho has convinced Morata that he will be a key player in his team next season as a replacement for Zlatan Ibrahimovic.
The IBTimes report that United are hopeful of concluding the deal before the weekend.
That would mean Morata is set to become the club's second signing of the summer.
Sweden international defender Victor Lindelof is the only arrival so far.
Meanwhile, Real team-mate James Rodriguez also wants to join United.
Rodriguez has threatened to stay at the Bernabeu if he is not allowed to move to Old Trafford.
