Wilshere, 17, has been compared to none other than Dennis Bergkamp and Liam Brady by Arsene Wenger, who has used the England Under-21 international sparingly as he seeks to nurture his academy's star graduate.
Despite becoming the youngest player to represent the club in the league last year, and impressing in the Emirates Cup in pre-season, Wilshere has made only one substitute appearance in the Premier League this season.
Championship and SPL clubs have shown an interest in signing Wilshere on loan but Wenger has admitted that if the player does leave Emirates Stadium, he will in all likelihood join a Premier League side, leaving Burnley in an advantageous position should they follow up their interest.
"He is a quality player, a young player who I believe will go on to be real star in the future," Coyle said. "If Arsene says he is available on loan then I think there would be a host of clubs interested in that type of player.
"We are interested in a lot of players to enhance the squad and if Jack Wilshere is one of those, who knows?"
Wenger admitted the idea was floated by Burnley manager Owen Coyle before the 1-1 midweek draw at Turf Moor.
Wenger said: "It is possible, with one condition, that he plays. Wilshere needs to play. Burnley is a good club which I respect. They play good football, but Jack must want to do it. I like the idea, but at the moment we are a bit short. Jack is in the squad nearly every game.''
Meanwhile, Arsenal have confirmed that another English youngster, Jay Emmanel-Thomas, has signed a new long-term contract.