The Gunners made the offer late on in the afternoon with the money they had left over after spending £56m on Borussia Dortmund striker Pierre-Emerick Aubameyang.
But it was not nearly enough to tempt Albion into selling and the conversation between the two club's decision-makers was brief.
Evans has not been named in the squad for tonight's trip to Manchester City, but that is because he picked up a hamstring injury towards the end of Saturday's 3-2 win over Liverpool at Anfield.
Jake Livermore, Kieran Gibbs, and Hal Robson-Kanu also miss out due to hamstring problems, which means Claudio Yacob and 19-year-old Sam Field start at the Etihad Stadium.
New loan signing Daniel Sturridge in on the bench but there is no place in the squad for new defender Ali Gabr.
There has been speculation over Evans's future throughout the window, but Manchester City decided to buy Aymeric Laporte from Athletic Bilbao instead and Arsenal focused their funds on Aubameyang.
The Northern Irishman has 18 months left on his current contract, but has been offered a new one worth £100,000-a-week.
He will now see out the season with the Baggies.
Earlier today, three youngsters headed out on loan. Defender Kyle Howkins joined Port Vale, left-back Max Melbourne went to Ross County, and goalkeeper Jasko Keranovic joined Kilmarnock.