Marca says the Premier League club have tabled an improved bid of €75m (£66m, $85.1m) for the striker which comes close to the 12-time European champions' asking price of €80m (£70.2m, $90.7m).
The Red Devils manager Jose Mourinho has been constantly in contact with Morata in the last few weeks. The Portuguese tactician gave the 24-year-old his first team debut when the former was at Real.
Mourinho has assured Morata that the situation will be resolved, which will then see him reunite with his former manager at Old Trafford. The striker, who scored 20 goals across all competitions, has also expressed his desire to play under United manager.
IBTimes UK revealed earlier this week that there had been a significant breakthrough in talks between Real and United over the weekend, with a deal expected to be completed in around 72 hours.
Marca now insists that the 20-time English champions are just "hours away" from agreeing a fee with the Champions League and the La Liga winners for Morata.
IBTimes UK already reported that Mourinho wanted to finalise the deal before the weekend. The 54-year-old manager was keen on the Spain international joining his new teammates when they fly to the United States for United's pre-season tour.
United have lost Zlatan Ibrahimovic, their last season's top goalscorer across all competitions. The Swedish striker suffered an injury in April and is expected to be on the sidelines for the rest of this calendar year.
Ibrahimovic was released earlier in the summer after his contract expired. Wayne Rooney is also heading towards the Old Trafford exit as his former club Everton have emerged as the front-runner for his signature.
Mourinho wanted to strengthen his strike force for the 2017/18 season, especially after Ibrahimovic's injury setback and Rooney's imminent departure. Morata is likely to take the former Paris Saint-Germain star's place in the English club's squad.