"I won't discuss our negotiations publicly, but I'm optimistic that something can be done," Gaine said Friday at his pre-draft news conference.
Clowney is scheduled to play on his fifth-year option in 2018, when he will get a significant raise because he was the No. 1 overall pick in 2014. Clowney made $7 million in 2017, and in 2018 he is scheduled to make more than $12 million.
"[I like his] height, weight, [and] speed," Gaine said. "The production, the performance, the pass-rush ability. The guy's a disruptive player. He plays the run well. So we're doing our best to fortify our nucleus of players, and we'd like him to be a part of it."
Clowney dealt with injuries during his first two NFL seasons; he played in his first 16-game season in 2017. Although Clowney's 15.5 sacks are tied for 23rd among NFL players since the beginning of the 2016 season, he has 37 tackles for a loss, which is second to only to Arizona Cardinals defensive end Chandler Jones.