Talks have been ongoing for the past couple of weeks about the Northern Ireland international, with City having had two offers turned down for a player that West Brom say is not for sale.
Evans, who had nine years at Manchester United before joining Albion in 2015, has two years left on his deal.
West Brom have also 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 while making 198 appearances.
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 for their captain.
City manager Pep Guardiola said last month he was still hoping to sign another central defender before the transfer window closed.
His fourth choice in that position is Eliaquim Mangala, who is not thought to have any long-term future at City.
Former Everton winger Pat Nevin speaking to BBC Radio 5 live:
Everyone knows Manchester City have a weakness at centre-back. Virgil van Dijk would be the perfect fit but if you don't get him then Jonny Evans, at 29, is probably coming in at his peak.
He's comfortable in a two, comfortable with three at the back, and he's not bad on the ball either. I am struggling to find a negative.
Just look at the Premier League at the weekend, centre-backs are really hard to find. People say he's not worth it, but there are so few of them around - so of course he is worth it.
Manchester City have had an £18m bid for defender Jonny Evans rejected by Premier League rivals West Brom.