The Red Devils have been in talks over a potential deal for the Netherlands international since the end of May, with the team's new head coach Erik ten Hag identifying the midfielder as a priority transfer target.
A recent report claimed that the deal had stalled, with De Jong's agents Ali Dursun and Hasan Cetinkaya telling Barcelona that their client has no intention of leaving this summer.
According to Italian journalist Fabrizio Romano, Man United have now come to an agreement with the Catalan giants over a deal for the Dutchman.
Romano claims that the package is worth €85m (£72m), with the 20-time English champions prepared to pay €75m (£63.5m) plus add-ons to bring the former Ajax youngster to Old Trafford.
However, the journalist claims that personal terms are 'still the issue', as the midfielder is keen to remain with Barcelona this summer despite their desire to sell him.
"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 it is understood that he would receive a huge increase in his wages if he stayed with the club this summer.
Last term, the midfielder made 46 appearances in all competitions, contributing four goals and five assists in the process, and he is regarded as a key player by the Catalan outfit.
However, Barcelona have accepted that they will need to make a big sale this summer, with the club looking to sign both Leeds United's Raphinha and Bayern Munich's Robert Lewandowski in the near future.
When asked about De Jong earlier this week, Man United boss Erik ten Hag told reporters: "I never talk about players who are not under contract of Man United, players from different clubs, so I cannot tell anything about this issue."
Barcelona president Joan Laporta said 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."
The Catalan outfit have now reportedly now agreed to sell the midfielder, but the deal hinges on whether De Jong is prepared to accept personal terms at Old Trafford.
The Dutchman has represented Barcelona on 138 occasions since arriving in the summer of 2016, contributing 13 goals and 17 assists in the process.