Asik has requested a trade on two occasions this season since finding himself slipping down in the pecking order behind Dwight Howard, who arrived from the Los Angeles Lakers before the start of the current campaign.
"We pushed to trade Omer in December," Morey told reporters. "We felt we had to make a fair and aggressive effort to do that. Obviously, he'd prefer to be a starter.
"At this point, Omer is very likely going to be here until the end of his contract at the end of next season, not this season.
"The window to trade him was [in December] and teams weren't aggressive enough to get him, so we're excited about him being a part of our future."
Asik hasn't played a game since December 2.