The Red Devils are believed to have already agreed personal terms with the Serbian over a £50m switch to Old Trafford.
But The Sun claim no move will be finalised until the clubs sort out extra payments United will have to make based on their results and the number of games Matic plays.
Jose Mourinho is a big fan of the 28-year-old and is keen for a reunion with him ahead of the new season.
Matic did not travel with the rest of his Chelsea team-mates on their pre-season tour of the Far East and has been training alone.
United are hopeful they can finalise a deal this week and add the midfield general to their ranks.
The reports claim both Antonio Conte and Roman Abramovich have signed off on letting Matic join their Premier League rivals.
The Blues have already added Tiemoue Bakayoko to their ranks who would be a suitable replacement.
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Matic would become United’s third signing after adding Victor Lindelof and Romelu Lukaku this summer.
Starsport understands the Red Devils are also closing in on a deal for Inter Milan winger Ivan Perisic.
United start the new Premier League campaign at Old Trafford against West Ham on Sunday, August 13.
