It is also understood that Ramos has told Madrid he wants to leave the club and join Louis van Gaal’s side.
Widespread reports in the British and Spanish media have linked the defender, who has two years left on his contract at the Bernabeu, with a move to Old Trafford.
Ramos has a buy-out clause of £142million but believes United’s offer represents good value for a player of his age and considering he has two years left on contract.
Madrid will attempt to appease Ramos over his contract but his brother and agent Rene is understood to have intimated to the club hierarchy that the opportunity for that has passed.
Ramos wants the move to Old Trafford this summer and says United are the only club he wishes Madrid to negotiate with.
The signing of such a Real Madrid talisman will be a major coup for United’s executive vice chairman Ed Woodward as he looks to strengthen Louis Van Gaal’s squad for their return to the Champions League next season.
They remain confident of signing Morgan Schneiderlin too from Southampton although a fee has yet to be agreed for the 24-year-old France international.