The Scottish-born American international joined the Trotters in January from MLS side Houston Dynamo and has only featured in three games since but has impressed his boss.
Coyle told The Bolton News: “He came to the club with the option of another year.
“Like everyone, I have been really impressed with Stuart on and off the pitch since he came here. The players really like him, and he's been terrific in each game he has played in for me.
“We had a meeting with his representative last week and, needless to say, I’d love him to be a part of this football club for a long time to come.”
After starting against Tottenham Hotspur and Wolves in February, the 24-year-old sustained an injury and only returned as a substitite in the final game of the season against Birmingham City.
However, Coyle believes there is still much more to come from the American, who played for his country against Czech Republic in Connecticut on Tuesday.
“You saw what the fans thought of him when I brought him on against Birmingham with half an hour to go," Coyle continued.
“I thought Stuart was excellent in those early games, and bounced back so well after getting the injury. It tells you a lot about him.”
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