The 35-year-old confirmed this week that he will be leaving the Santiago Bernabeu after 16 years, and several clubs have been linked with a move for the veteran defender, including United, Manchester City, Paris Saint-Germain.
Ramos made 671 appearances across competitions for Los Blancos and won 22 trophies, and will be leaving the club on a free transfer after failing to agree on a new deal.
According to the Daily Mail, the Red Devils have removed themselves from the race as executive vice-chairman Ed Woodward and the board remain unhappy with Ramos and his brother Rene's conduct towards them the last time they were in talks to sign him.
Ramos admitted in 2015 that he nearly made the switch to Old Trafford. He told COPE: "There was an offer from United. I considered changing for a while but Real Madrid was always my priority and there were no economic reasons.
"Did I feel cheated [by Madrid]? No, but I did feel disappointed for a while. There were some troubles that came to an end. Flattery from big clubs is always good as this is because you are doing well, but the fans and my club want me."
Pep Guardiola's City are seemingly leading the race to sign Ramos as the Spaniard, who is a huge fan of his fellow compatriot, is keen to bolster his defense ahead of the new season. United are also looking to bring in a new partner for Harry Maguire, and Ramos' Madrid team-mate, Raphael Varane, and Villarreal star Pau Torres, have emerged as two targets for Ole Gunnar Solskjaer.