The 28-year-old has a verbal agreement with the Red Devils.
United now need to agree a fee with Chelsea for the transfer to move forward, according to the Manchester Evening News.
Recent reports have said the Blues will only let Matic go when they secure a deal for Monaco star Tiemoue Bakayoko.
United had also been considering a swoop for Tottenham midfielder Eric Dier.
Matic enjoyed a successful season at Chelsea last time out playing alongside N’Golo Kante.
The Serbian helped Antonio Conte’s side to the Premier League title and an FA Cup final.
Matic made 40 appearances in all competitions for Chelsea last season.
Jose Mourinho previously worked with Matic at Stamford Bridge and thinks he is the perfect man to bolster his midfield.
United have been busy this summer already, signing defender Victor Lindelof from Benfica.
The Red Devils have also been linked with Real Madrid striker Alvaro Morata and Inter Milan midfielder Ivan Perisic.
It is the third player that Manchester United have agreed terms with after Fabinho and Morata, according to reports. Surely English media has no chill when it comes to Manchester United.
Mourinho’s side start the new Premier League campaign with a home match against West Ham.