There have been suggestions that the English club's failure to qualify for next season's Champions League will make it almost impossible for them to convince him to join the club.
However, according to journalist Florian Plettenberg, the Red Devils are refusing to give up in their pursuit of the Dutchman, who is allegedly available for the right price this summer.
Plettenberg claims that the 20-time English champions are 'still pushing' for the midfielder, but 'it will not be easy' to bring him to Old Trafford in the upcoming market.
De Jong was again a vital player for Barcelona during the 2021-22 campaign, making 46 appearances in all competitions, contributing four goals and five assists in the process.
Meanwhile, he has now represented the Catalan outfit on 139 occasions since arriving from Ajax in 2019, providing 13 goals and 17 assists.
De Jong still has more than four years left to run on his contract at Camp Nou, but there have been claims that he will be available this summer, as Barcelona look to raise funds to bring in Bayern Munich's Robert Lewandowski.
The La Liga giants have managed to ease their financial problems, but there are still off-the-field issues at the club, which will again restrict them in this summer's market.
"We will do what is best for the club. If the economic levers that we want to activate come off, no player will leave for economic reasons," Barcelona president Joan Laporta recently told Catalunya Radio when questioned on a possible exit for the midfielder.
"I insist. Let the sports management work. There are players who are quoted to have received offers. In the sports area, Xavi and Mateu Alemany rule and he ends up deciding the club's sports management. I will follow what the professionals of the club say."
Meanwhile, Barcelona head coach Xavi also recently conceded that the former Ajax youngster could leave Camp Nou this summer for financial reasons.
"He's an important player for me, and he started every game except for a few due to rotations. But it depends on the economic situation. For me, Frenkie could be one of the club legends," Xavi told reporters.