The Madrid captain is yet to renew his contract at the Bernabeu, which expires on June 30, and has been free to negotiate with other clubs since Jan. 1.
City are aware that a deal would be complicated but are looking to reinforce their defence ahead of next season.
Sources told ESPN that Pep Guardiola is a fan of Ramos and would be keen on a deal for the veteran centre-back.
Ramos will turn 35 in March but City's assessment is that he is still able to perform at the highest level and would be an "interesting" option to provide leadership and experience at the back.
Sources told ESPN that City would have no problems taking on Ramos' substantial salary demands -- which have proved a stumbling block in Madrid -- given that he would arrive on a free transfer.
Nonetheless, sources told ESPN in November that Ramos' priority remains renewing his Madrid contract and talks began in the autumn.
Ramos would like a two-year extension, and Madrid are willing to accept that, making an exception to their usual one-year policy for over-30s.
The situation has dragged on longer than expected, though, due to the impact of the coronavirus pandemic on the club's finances.
City are aware, too, that the most likely outcome is that Ramos will eventually renew, but want to be well-positioned if negotiations break down and a departure from the Bernabeu becomes a realistic possibility.