The Northern Ireland international left City's local rivals Manchester United and completed a switch to the Hawthorns in 2015. He has established himself as one of the best defenders in the league with the Baggies, despite their recent struggle.
According to The Sun, West Brom are braced for a "firm bid" from Pep Guardiola's side for Evans in the next day or so. Alan Pardew's side are expected to sanction the sale of the centre-back as they are aware it will be difficult to keep him at the club.
Evans was subjected to interest from City and Arsene Wenger's Arsenal at the end of the last summer transfer window. Both teams failed to agree to a fee with West Brom, which resulted in the 30-year-old continuing in the West Midlands.
He has less than 18 months left on his contract at West Brom. The report also claims the Gunners are also believed to be interested in signing the former United defender in the ongoing window.
Arsenal are refusing to make a concrete approach for the central defender, which could see them lose out in the race to sign Evans. This should also give City a clear path in signing the defender, who has also been linked with a move to his former club.
The Mirror earlier reported that Jose Mourinho's side are keen on re-signing Evans in January. However, it looks unlikely that Jose Mourinho's side will make an approach for the relegation-threatened side's centre-back.
Pardew has already admitted it will be difficult for West Brom to keep Evans if they receive an "acceptable" offer for his signature.
"That name's not entered into my thinking, or names at Arsenal or Manchester City or Manchester United if I'm honest. When there is an offer on the table we will exercise our plan if the offer is acceptable to us," the former Newcastle United manager was quoted as saying by Sky Sports.
West Brom are 19th in the table with 19 points after 23 games. Retaining Evans will be vital in their efforts to avoid relegation to the Championship, with 15 more games left until the end of the season.