City saw an £18 million bid for Evans rejected last week by West Brom with Pep Guardiola keen to bring in the Northern Ireland international to bolster his defensive options.
Sources have told ESPN FC that City are increasingly confident of landing Evans due to the player's desire to make the switch and West Brom boss Tony Pulis refusing to rule out a move after the initial bid was rejected.
Guardiola is an admirer of Evans and is keen to sign a new centre-back before the close of the transfer window.
Evans joined West Brom in the summer of 2015 from Manchester United and has quickly established himself as a key player at The Hawthorns.
West Brom signed Evans for a fee in the region of £8m and they would be looking to make a big profit on him if they are to sell him to City.
West Brom are reluctant to lose Evans, who has been made captain this season, but they are aware that the player is interested in a move to City.
Evans still lives in Manchester and travels to West Brom for training every day, and City are able to offer the defender the chance to return to playing in the Champions League and the opportunity to challenge for trophies like he was doing at Old Trafford.
West Brom are already looking at defensive reinforcements in case Evans does leave and have made a bid for Tottenham centre-back Kevin Wimmer.