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.

Mark Mason, 48, from Rhyl, died after being stabbed at the town's Home Bargains car park on 27 October 2016.
James Davies, 20, from Liverpool, denies murder and malicious wounding with intent.
Anthony Baines, 30, and Mark Ennis, 30, also from Liverpool, deny the same charges.
Jake Melia, 21, also from Liverpool, has admitted all charges.
The trial at Mold Crown Court has heard the attack was the result of a turf war which had erupted between two rival gangs over the control of the drugs trade in the Rhyl area.
On Monday, Mr Davies told the court he approached Mr Mason's car with a knife and banged its handle on the car window.
He said: "I just wanted them to see it and be scared.
"It was a stupid idea when I think of it but I didn't know it was going to end in the way that it did."
He said he did not inflict any injuries and he did not see how Mr Mason suffered 22 stab wounds.
He added he only discovered Mr Mason had been stabbed when he was told as they drove away.
The case continues.

Monitor is looking into the financial sustainability of Southend University Hospital Foundation Trust "on behalf of patients", the NHS regulator said.
Finances became a concern when a planned £7.8m deficit grew, for which Monitor could see no recovery plan.
The hospital blamed the larger-than-anticipated deficit on growth in demand and increased staff recruitment.
Hospital chairman Alan Tobias OBE, said:  "The overspend is a result of vital investment in services and more staff to ensure high standards of patient care as well as the rise in attendances.
"We welcome this review to clearly demonstrate to Monitor - as well as patients and stakeholders - our future financial plans are both robust and sustainable."
The hospital has pledged to cut the deficit while maintaining "high quality care for local people" and said it was committed to returning to a surplus within three years.
Monitor said its investigation will look into the state of the hospital trust's finances, assess the strength of its financial management and explore ways of improving its sustainability, on behalf of patients across the region.
The investigation will also examine how the trust works with other local health and care organisations to respond to the financial challenges it faces.
Katherine Cawley, regional director at Monitor, said: "People are relying on Southend to provide them with quality healthcare now and in the future, and we are concerned that this may be at risk because of the trust's poor financial performance.
"We have launched this investigation to find out more about the financial situation at the trust and to establish what can be done to improve things."
Manchester City have had an £18m bid for defender Jonny Evans rejected by Premier League rivals West Brom.
[NEXT_CONCEPT]
A drug dealer has admitted waving a knife around in a Denbighshire car park before a man was fatally stabbed but denied using the weapon.
[NEXT_CONCEPT]
A £9.8m deficit recorded for 2014/15 at an Essex hospital is to be investigated by a health service watchdog.