Barry was entering the final year of his contract at Goodison Park, and was facing a season on the bench due to competition for places.
"I was happy at Everton and prepared to do my last year which included playing in European football this season," the Birmingham Mail quotes Barry as saying. "I was told by the club there was interest from a couple of Premier League teams which got me thinking a bit.
"I had some great conversations with [Everton manager] Ronald Koeman. He is a very honest manager and we discussed everything at length and he wanted to keep me around.
"Did I want to stay at Everton? Which was very comfortable for me, but I realised I wasn't going to be first choice as they were signing a lot of players. Was I ready for another challenge? So that's where my head was.
"I heard of West Brom's interest, thought long and hard about it and realised that was what I wanted. I am not the sort of player to sit back and be comfortable. I was really ready for this new challenge."
Barry has made 628 appearances in the Premier League in his career so far.