The two clubs are in talks over a deal for De Jong, who has given United enough encouragement that he would be willing to move to Old Trafford for Erik ten Hag to believe an agreement is possible.
According to sources, Barcelona have been looking to receive €100 million, but United's valuation is closer to €60m with add-ons for the Netherlands midfielder.
Sources have told ESPN that the feeling within the club is that Barca are actively trying to offload De Jong in an effort to raise funds to facilitate their transfer plans for the summer and that it should be reflected in the cost.
Ten Hag and football director John Murtough are prioritising the signing of a central midfielder over forwards and defenders, but there are other names on the list beyond De Jong.
Sources have told ESPN that discussions with Barca have been accelerated after De Jong intimated he would be open to a move to United.
The 25-year-old initially appeared reluctant to leave Barcelona for a club outside the Champions League, but that stance has softened.
Paris Saint-Germain also showed an interest when Barcelona offered De Jong to Europe's major clubs. But as things stand, United are the most viable option.
Sources have told ESPN that United ultimately decided they did not want to get into a battle with Liverpool and will instead turn their attention to other targets.
There is interest in Ajax winger Antony, whom Ten Hag knows well from their time together in Holland, and RB Leipzig striker Christopher Nkunku, although the priority remains to secure a central midfielder.
After the departures of Nemanja Matic and Paul Pogba, Ten Hag is short of numbers in midfield while the budget to strengthen the attack is likely to depend on whether a club would be interested in Anthony Martial.