The 25-year-old's future has been up in the air all summer, with new United boss Erik ten Hag keen on a reunion with his former player at Old Trafford.
The Catalan side appreciate the quality of the Dutchman but remain in a perilous financial situation as things stand, leaving them open to the sale of one of their most prized assets.
After several weeks of negotiations, a fee of around £73m was finally agreed between the two clubs for De Jong on Thursday, but up to this point, the midfielder has turned down a move to the Premier League.
However, the Red Devils are prepared to offer improved terms in the hope of persuading De Jong to change his mind, according to Spanish outlet Sport.
The report reiterates De Jong's desire is to remain at the Camp Nou for the 2022-23 campaign, despite the club's best efforts to push him towards the exit door.
However, it is also stated that following the midfielder's inclusion within the squad for the pre-season tour of the United States, which flew out on Saturday, Barca remain keen not to force the issue too much.
Meanwhile, whilst the likes of Memphis Depay will also take part in the training camp abroad, despite the possibility of a move away from the club this summer too, Barcelona boss Xavi was refused entry to the US on Saturday due to issues over documentation.