It is understood talks have taken place over the past couple of weeks but there has been no agreement over a fee for the Northern Ireland international.
Evans, who played 198 times for Manchester United before joining Albion in 2015, has two years left on his deal at The Hawthorns, where he is captain.
West Brom have turned down a £10m bid from Leicester for the 29-year-old.
Evans joined United as a schoolboy before going on to win three Premier League titles and two League Cups.
He joined West Brom after being told he did not figure in former United manager Louis van Gaal's plans.
Twelve months ago, Arsenal showed an interest in signing Evans but West Brom rejected the approach.
City manager Pep Guardiola said last month he was still hoping to sign another central defender before the transfer window closed.
His fourth choice option is Eliaquim Mangala, who is not thought to have any long-term future at City.
It is easy to see why Guardiola feels Evans would fit into his squad as the additional central defender he has been wanting.
The Northern Irishman is good with the ball on the floor, is fast and an excellent reader of the game. His injury record might be a concern but, equally, he still lives in the Manchester area so settling in would not be a problem and he knows the Premier League.
His background as a United fan and player might be an issue for some but, given an approach from Arsenal was rebuffed 12 months ago, he is evidently viewed across the game as a player capable of performing at Champions League level.