The 29-year-old Spain international has made up his mind to leave Bernabeu this summer after Real allowed his contract to run into its final two years and has asked the club to consider United’s offer and begin talks.
The discussions with Ramos and his representatives have encouraged the Red Devils to formally launch a bid, the reports states.
United believe that they can benefit from the Ramos’ contract dispute with Real to secure his services.
Ramos has a £143m buy-out clause in his current contract, but it is believed that Real would demand closer to £46m for the defender.
Ramos, who currently earns £3.9m a year, is said to be demanding a new £13.5m a year deal, which Real are reluctant to sign.
The Spaniard had joined Real from Sevilla for €27 million in 2005 and has played more than 400 games for the club.
He has also won 125 Spain caps since making his full debut in 2006 and represented the country in three FIFA World Cups and two European Championships.
United goalkeeper David De Gea is linked with a move to Real, however, Red Devils’ initial offer for Ramos is believed to be independent of any deal involving him.
United had turned down Real’s initial bid, thought to be in the region of €18m, for the shot-stopper and but could use him as a makeweight in their second offer for Ramos if Real reject their first bid.
De Gea spent four seasons at United after joining the club from Atletico Madrid for £17.8m and has one year left on his contract at Old Trafford.