The Spain international has been linked with Old Trafford since stories emerged in June that he had decided to put an end to his career at the La Liga giants after growing frustrated with the club's president over the unsuccessful negotiations to get a new improved contract.
Louis van Gaal appeared to be open to the possibility of his signing, saying: "Maybe Mr Ramos is in the process, you never know."
Spanish newspaper El Pais reported that the Premier League big dogs launched an improved €60m (£42.6m, $66.15m) bid to secure the signature of the 29-year-old defender.
Marca then claimed that United asked Ramos to make a public statement – equivalent to a transfer request – expressing his desire to leave Real Madrid.
But Marca now claims that the saga is set for a sensational U-turn after a crucial meeting during Real Madrid's pre-season tour in China between Ramos, his agent, club chairman Florentino Perez and club general director Jose Angel Sanchez.
Marca says that following the meeting Ramos is "much closer to renewing his deal at Real Madrid than to a move to United".
The Spanish publication added that even though a new deal is yet to be reached all four parties agreed on principle.
Marca claims that Real Madrid are ready to improve Ramos's current deal, but the player is also happy to reduce his original salary demands as a sign of his commitment to the club.
The Spaniard earns €6m a year after taxes but the negotiations for a new deal stalled after he demanded €10m – with Los Blancos unlikely to go further than €7.5m.
AS claimed that United offered him a deal worth €55m over the next five years to move to Old Trafford but Marca now suggests the centre-back is ready to snub the Red Devils' lucrative offer to stay at Real Madrid instead – even though he will earn less.