Coach Pep Guardiola values Ramos' experience and leadership qualities.
City are looking at offering him a tailored deal based on the player's form and own priorities: a two-year contract, with the possibility of a third if performance allows, or the alternative of a spell in Major League Soccer at City Football Group's New York City FC.
Ramos' contract at the Bernabeu expires on June 30 and his priority is still to renew with Madrid, but he will move on if an agreement with president Florentino Perez cannot be reached.
Sources told ESPN a meeting between the club captain and president is planned for the next few days when a definitive decision will be taken.
Negotiations have been stalled since the start of the season, and ESPN reported in January that City had taken an interest in Ramos' situation.
Guardiola believes a footballer with Ramos' qualities would increase the competitiveness and experience of the squad in such high-profile matches.
Young defender Eric Garcia is leaving to return to Barcelona and Nathan Ake played a limited role last season due to injuries -- making only 10 league appearances -- while Aymeric Laporte has gone from automatic pick to substitute over the last two years.
Ramos made just 15 La Liga appearances for Real Madrid himself, suffering a series of injuries which saw him left out of the Spain squad for Euro 2020 by coach Luis Enrique. However Guardiola believes that Ramos could play a Vincent Kompany-style role in the City squad.
Sources told ESPN that City's data analytics suggest Ramos' physical performance is akin to that of a much younger player, which is why the club would be willing to contemplate a two-year contract.
Ramos is aware of City's interest but wants to speak to Perez before taking a final decision.
He would like a two-year extension at Madrid, but so far the club have been unwilling to make an exception to their year-on-year policy for the over-30s.
Madrid have signed David Alaba to strengthen the defence ahead of next season, and it remains to be seen if Perez will concede and offer Ramos a deal that includes a second year, based on appearances.
The future of Ramos' centre-back partner Raphael Varane is also in doubt, with his contract expiring in 2022. Madrid would consider offers for the France international this summer if he chooses not to renew.