The Red Devils have allegedly reached a full agreement with Barcelona for the Netherlands international, but the deal is far from complete, as De Jong is still determined to stay at Camp Nou.
Italian journalist Fabrizio Romano recently claimed that the package is worth €85m (£72m) to Barcelona, with the 20-time English champions prepared to pay €75m (£63.5m) plus add-ons to sign the former Ajax youngster, which would allow the Catalan giants to register their new signings.
However, De Jong has often referred to Barcelona as his "dream club", and the Dutchman has travelled on the team's pre-season tour of the United States.
"I prefer to stay with Barcelona. Barcelona is just my dream club. Also from a young age. I just said it too. I've never regretted my choice despite the fact that I had hoped for more in terms of prizes than I've achieved so far. But I have never regretted my choice," the midfielder told ESPN earlier this summer.
"You are always flattered when teams show interest in you as a player," De Jong also told reporters in June. "But I think I am at the biggest club in the world at the moment and feel fine there so there is no news."
De Jong also still has another four years left to run on his contract at Barcelona, and he was a key player for the club last season, making 46 appearances in all competitions, contributing four goals and five assists in the process.
According to Sport, the midfielder would prefer to join Bayern this summer, as he is desperate to continue challenging for major honours at this stage of his career.
The report also claims that the prospect of reuniting with his close friend, Matthijs de Ligt, is appealing, with the Juventus defender believed to be nearing a switch to Allianz Arena.
Man United's failure to qualify for next season's Champions League is believed to be a huge issue for De Jong when it comes to convincing him to make the move to the club.
However, a deal to take De Jong to Bayern would reportedly be complicated due to the poor relationship between the two clubs, with the German outfit unhappy with how Barcelona acted during the negotiations for Robert Lewandowski.
The Dutchman has represented Barcelona on 138 occasions since arriving from Ajax in the summer of 2016, contributing 13 goals and 17 assists in the process.
He remains a key player for the Catalan giants but is viewed as an individual capable of raising a high transfer fee, which would free up space to register their new signings and potentially add more players to the squad before the end of the window.
De Jong is also allegedly in line for a huge pay rise next season if he remains at Camp Nou, and he is not believed to be prepared to accept a reduction despite the recent rumours surrounding the situation.
Barcelona president Joan Laporta told reporters recently: "There are clubs that want him, not just United. We have no intention of selling him, he wants to stay. Frenkie is considered as one of the best midfielders in the world by all the experts. We are happy to have him."
Bayern have made three signings during this summer's transfer window, bringing in Sadio Mane, Ryan Gravenberch and Noussair Mazraoui.