United initially had a £52m bid rejected but are hoping a new offer of £60m will suffice for the Madrid board.
Jose Mourinho's side are in the market for a marquee forward after plans to bring Antoine Griezmann to the club were scuppered by a transfer ban.
The newspaper says that United's executive vice-chairman Ed Woodward will not be held to ransom by the Real hierarchy, who could make life hard for the Premier League club over their refusal to sell David de Gea.
As revealed by Telegraph Sport in April, United have made Morata their top target to replace Zlatan Ibrahimovic and are close to clinching his signature.
Although a deal for Morata may not prove straightforward.
While £60 million would still fall some way short of that enormous fee, it's a generous offer for a player who isn't a regular starter.
The Swede had a stellar first season at Old Trafford, but a serious knee injury sustained towards the end of the campaign has led to the club deciding against offering him an extension. "All that remains now is for the directors of the English team to negotiate a transfer fee with their Spanish counterparts".
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United manager Mourinho has lined up a swoop after losing out to Chelsea in the race for Everton's Romelu Lukaku.
Morata's agent recently spoke to Foot Mercato and did not deny that a move could be in the offing.
United need a striker who will at least score 20 goals in the league next season and Alvaro has never scored 20+ league goals for any team as yet. "I have not yet met with the representatives from Real Madrid".
Alvaro Morata has been linked with a Real Madrid exit. The player will take a clear and definitive decision in the coming days.
However, it has been claimed that he suffered a change of heart following winning the Champions League with Real Madrid and AC Milan are no longer in the chase for Morata.
Real Madrid striker Alvaro Morata is ready to join Manchester United this summer.
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