La Liga heavyweights Real Madrid were thought to be the favourites to sign the Paris Saint-Germain superstar but United have entered the race.
Neymar has been in the French capital for less than a year but is already said to be desperate to leave. He departed Barcelona in August in a deal that cost PSG almost £200million.
According to the Daily Mirror, Manchester United are preparing a stunning move for the Brazilian forward and are willing to pay a world-record fee and match his £30m-a-year salary to complete the deal.
It is claimed the Red Devils believe they have the upper hand over Real as they have a working relationship with Neymar Santos senior, a United fan who would love for his son to sign at Old Trafford.
Neymar has endured a turbulent year at PSG and Unai Emerys exit at the end of the season will not alter his desire to leave.
Real Madrid boss Zinedine Zidane has repeatedly played down rumours linking Neymar with a switch to the Bernabeu.
This week, he remained coy on Reals interest in the Brazil international but said he saw no issue with a potential partnership with Cristiano Ronaldo.
Good players are always compatible, he said. I remember one year they said I wasnt compatible to play with Djorkaeff. We won the World Cup together.
So good players can always play together. On the pitch there is a chemistry.
I dont know if we are negotiating for Neymar. Whats worrying us is what is going on at the moment.
I havent asked Neymar personally because I dont get involved in these things. We have to finish this season, the rest will come after.