United are set to pay €65 million up front for the midfielder, who is manager Erik ten Hag's top summer target.
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Performance-based add-ons, which could take the cost above €80m, are still to be agreed, as well as the player's personal terms, and sources close to negotiations have told ESPN the deal is not yet done and that there are still "significant" hurdles to overcome.
However, there is now increasing confidence at both Old Trafford and Camp Nou that an agreement can be wrapped up to make De Jong the first signing of Ten Hag's reign at United.
The Netherlands international is Ten Hag's key transfer target. There is hope he could be on the plane when the squad depart for their tour of Thailand and Australia on July 8.
Sources at Barca also acknowledge they are closer to reaching an agreement, but say the final figures are still to be determined, both in terms of the fixed fee and add-ons.
However, there is an acceptance that the total package won't reach the €100m they had initially requested when talks began between the to clubs.
The same sources add that they "don't believe" a potential transfer will be completed by June 30 (Thursday), which is when the club close their accounts for the 2021-22 campaign.
Barca still need to raise money this week to avoid ending the year with losses, but hope to do so through the sale of 10% of their LaLiga television rights.
De Jong has publicly and privately said he would prefer to stay at Barca -- as reaffirmed by former coach Ronald Koeman on Monday -- but could be forced to reconsider his position given the Catalan club's push to move him on.
Barca paid €75 for De Jong when they signed him from Ajax in 2019 with another €11m in potential add-ons.
Ten Hag worked with De Jong at Ajax and has been keen to bring the 25-year-old to United. Barcelona are also keen to let the player leave in an effort to overcome their financial problems and fund other transfer business this summer.
After completing their move for De Jong, United will turn their attention to signing a versatile forward who can play in a number of positions across the front line.
Brazilian winger Antony is on the list of potential targets, but so far United have been put off by Ajax's €80m valuation. United believe a fee closer to €40m is more in line with the 22-year-old's market value but Ajax have made it clear they are under no pressure to offload the forward despite the player encouraging any offers to be seriously considered.
Sources have also told ESPN that United are waiting to hear from Christian Eriksen about where he will play next season. United have been encouraged by Tottenham backing away from a deal, but Eriksen also has an offer from Brentford, where he spent the second half of last season.
ESPN's Moises Llorens also contributed to this report.