United made a bid of £55 million ($69.9m) plus add-ons on Friday, which was knocked by Chelsea.
It was their third rejected bid after previous offers of £40m ($50.9m) and £50m ($63.6m).
Chelsea, according to sources, have offered United a deal of £58m ($73.7m) plus another £7m ($8.9m) in performance-related bonuses.
They have also offered to meet United officials in person in the hope of resolving the situation.
Mount, 24, is keen to move to Old Trafford after ruling out the prospect of signing a new contract at Stamford Bridge.
He has one year left on his deal and Chelsea do not want to lose him for free next summer.
United believe their latest offer reflects a good deal for Chelsea given Mount is under contract for only another year and has told the club he wants to move.
Chelsea, however, have insisted that they have already significantly reduced their initial valuation of £70m ($89m).
Sources have told ESPN that United want to seal their move for Mount quickly to ensure the England international is on their preseason tour of the United States, which starts in New York on July 20.