Sources have told Goal that City agreed personal terms with Evans last week, and the Blues are now trying to broker a deal with West Brom.
It has been reported that the Baggies have already rejected two bids from City, one worth £18 million, though the Blues remain confident they will reach a breakthrough before the transfer window closes at the end of August.
Pep Guardiola has been in the market for an experienced, strong defender all summer, but has seen attempts to sign Virgil van Dijk and Leonardo Bonucci hit the buffers.
Van Dijk has handed in a transfer request at Southampton but City insist they will not go back in for the Dutchman, such is the Saints' lofty asking price of around £60m.
That has opened the door to a move for Evans, who joined West Brom from Manchester United for £8m in 2015 and could fetch around £20m.
A move for Middlesbrough's Ben Gibson had also been considered, though Evans is regarded as a better option than the 24-year-old at this moment in time.
Evans would be considered a replacement for Aleksandar Kolarov, who joined Roma earlier this summer, and therefore fourth-choice centre-back at the Etihad Stadium.
Northern Ireland international Evans, should he complete the move, would also boost City's homegrown quota for Uefa competitions.
Baggies boss Tony Pulis has previously insisted Evans would remain at the Hawthorns for next season, but cast doubt on that at the weekend by intimating he was not entirely convinced about the defender's injury, which kept him out of Saturday's 1-0 win against Bournemouth.
“Jonny felt his hamstring in training on Friday and we didn't want to take a chance," Pulis told reporters. "He went in to see the doctor at the training ground and came to the ground and told me on Saturday it was sore. So we take his word for it.”
Eliaquim Mangala will be allowed to leave City if Evans is signed.
The Frenchman has been training with the Blues in recent weeks and is currently with the first-team squad in Girona, but is only considered a temporary option while other options are pursued.
Similarly, 19-year-old Tosin Adarabioyo is expected to complete a loan move once a new recruit arrives.
City remain in the hunt for another forward, with long-term targets Alexis Sanchez and Kylian Mbappe still in the mix.
The Blues believe they would need Alexis to hand in a formal transfer request to force Arsenal's hand, though sources close to Guardiola insist he is not encouraging the Chilean to kick up a fuss.
The Chilean informed Arsenal of his desire to leave the club and join City at the start of July, but has so far not taken similar action to Philippe Coutinho, who is attempting to force through a move to Barcelona from Liverpool.
City have been concerned about interest from Paris Saint-Germain in both Alexis and Mbappe, though sources at the French club have told Goal France that they will not pursue a move for the latter this summer.
Mbappe was the subject of a €40m bid from City last summer and the Premier League club will fight to secure his signature this summer until he formally commits to another club, or signs a new deal at Monaco.
Mbappe has told City he would join them this summer if they can reach a deal with Monaco, though the two clubs are well apart on their valuation of the 18-year-old, with the French champions holding out for €180m.
The teenager is also open to joining both PSG and Madrid.
With PSG needing to sell players before stumping up the cash for Mbappe, and Real Madrid keen to sell one of their high-profile forwards, most likely Gareth Bale, before they will make a move, City have been presented a window of opportunity.
They too, however, would need to sell off Samir Nasri, Wilfried Bony, Jason Denayer and Eliaquim Mangala to boost their own coffers.