United manager Jose Mourinho wants the Serbian to become his third signing of the summer.
Victor Lindelof and Romelu Lukaku are the two names to join so far - and Chelsea ace Matic is next on the club's shopping list.
The Times report an opening £35m bid has been rejected.
And United could be forced to up their offer by £15m to get a deal done.
Roman Abramovich has approved a sale of Matic to the Manchester giants this summer.
The Chelsea owner will allow the 28-year-old to join a direct rival as a reward for his years of hard work for the club.
But he wants £50m in order for a transfer to go through.
The Times report Chelsea are imposing a 'Lukaku tax' on United after they pinched the Belgian striker from under their noses.
Romelu Lukaku was signed by the Old Trafford giants for £75m earlier this month despite the Blues agreeing personal terms with the player.
Chelsea were forced to look elsewhere for a new striker, with Alvaro Morata instead joining the club.
He completed a move yesterday.
Despite securing two signings already, Jose Mourinho is not happy with Man Utd's transfer business so far.