Baggies boss Alan Irvine, who worked with the New York Red Bulls star at Goodison Park, asked Cahill about his Australia teammate.
Irvine said: "I spoke to Tim Cahill about Jason and he recommended him very, very highly both as a player and character."
Davidson, who has signed a two-year, also revealed: "I'm very close to Timmy and during the World Cup I had other offers on the table but he said from day one that he wanted me to go to the Premier League.
"When West Brom came along and negotiations started I told him about that and he was with the coach at Everton and he only had great words for him. He said it was a very good club with a lot of potential that gives youngsters a go."
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