The 30-year-old, who has spent the past two years in East London, admitted earlier this week that he wants assurances that he will be a regular next season.
Adrian has been made to battle it out with Darren Randolph for a starting spot, being used in fewer than half of the Hammers' Premier League games.
Bilic has triggered the clause inserted in the Spaniard's deal, but Alvaro Arbeloa - a free signing upon leaving Real Madrid last summer - will not be offered fresh terms.
"Adrian is out of contract but we have an option, and we activated that option. He is our player and we are happy with him," Bilic told reporters.
"Arbeloa is a different situation. He came for one season and his contract is out now. But of course I am going to discuss those things with the players, not with the press.
"We don't have to improve the squad. We don't need squad players. We want to improve the quality in some of the positions. I'm not going to talk about names now.
"Most players are linked with West Ham in the papers. To be fair, the big majority of those rumours, there is no truth in them. But we have our targets and we are going to try to do them."
Arbeloa, who has previous Premier League experience with Liverpool, featured just four times for the Hammers in all competitions this season.