Their new bid is yet to be finalised but sources suggest it is expected to be between £55 million ($69.9m) and £60m ($76.2m).
Erik ten Hag's side have had two formal bids for the player rejected in recent days, the latest totalling £50m ($63.5m).
Chelsea were originally holding out for £70m ($88.9m). Although sources suggest that the club are softening their stance over that valuation -- they responded to United's last bid with a £65m ($82.6m) counter-proposal -- it remains to be seen how much they are willing to compromise in order to complete the deal.
Mount wants to move to Old Trafford and is hopeful an agreement can be reached. He has one year remaining on his contract at Chelsea, where talks stalled over an extension several months ago.
Chelsea are under pressure to complete deals before June 30 to aid compliance with UEFA's Financial Fair Play regulations having spent more than £650m ($825.6m) on players in the first full tax year of Todd Boehly and Clearlake Capital's ownership.
That situation has eased, however, with several outgoings set to be finalised including Kai Havertz' transfer to Arsenal, Mateo Kovacic joining Manchester City, as well as Kalidou Koulibaly, Edouard Mendy and Hakim Ziyech moving to clubs in Saudi Arabia.