The Red Devils made a third bid worth an initial £50million plus a further £5million in add-ons for the England midfielder.
However, that proposal has been turned down by Chelsea.
United had been insistent that their third offer was final and they were prepared to walk away from the deal if Chelsea knocked it back.
But with Erik ten Hag identifying Mount as a key signing for the club, United sources remain hopeful of coming to an agreement.
Despite Chelsea playing hardball over the fee, they are willing to let Mount join their Premier League rivals and have offered to meet with United in order to find an amicable solution.
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A deal totalling £60million would likely be enough to get it done.
Mount has made it clear that he wants to complete a move to Man United this summer.
The 24-year-old has just 12 months remaining on his contract at Stamford Bridge.
But Chelsea are keen to get their full worth for the academy graduate.
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Mount has scored 33 goals in 195 appearances for Chelsea since making his first-team debut in 2019 and was a key part of their Champions League success in 2021.
He has also won the UEFA Super Cup and Club World Cup with Chelsea and has been capped 36 times for England.