The Gers are believed to have rejected a £3m bid from the Baggies for the defender before their 3-1 win over Maribor in the Europa League qualifying rounds last night.
Liverpool legend Gerrard was disappointed by West Brom's advances and has hailed the skipper for remaining focused ahead of the game.
"James was magnificent," he said in a post-match press conference. "Good players get attention from other clubs. I don't appreciate that attention 24 hours before a big game. I'd like a phone call off of big Darren [Moore, West Brom manager] at some point.
"But James handled it very well and that was probably his strongest performance so far. He's got ability, he's got power - there is more to come from James. He still has gears to go into. For me, he's a Premiership footballer. He can play in the Premier League no problem.
"James knows what I think of him. It was a big decision to make him captain of this club. I speak to him every day. It's up to James whether he tells you what has been said between us over the past few days but he has my full support. I want James Tavernier here as long as I'm here."
The summer transfer window has now closed for permanent signings in England, although EFL clubs can still bring in players on loan deals until the end of the month.