The Express & Star says the Baggies rejected an opening £18 million bid for the Northern Irishman just under two weeks ago but reports from Manchester suggested Pep Guardiola was readying a second bid north of £20m to test the club's resolve.
However, Albion – who value Evans around £30m – are yet to be contacted with another offer from City.
Evans has missed the opening three games of the season with a hamstring problem but the 29-year-old has been named in Northern Ireland's squad for the World Cup qualifiers that start next week.
“I don't want to comment too much on Jonny's fitness because it's between him and his club at the moment," said Northern Ireland boss Michael O'Neill. “In conversations (I've had) with Jonny, he's optimistic he's going to be fit to play. We have to see what happens at the weekend."