Wilshere was not part of the Arsenal squad that won 2-0 at the Reebok Stadium on Sunday and a hamstring problem means he is a doubt when the two sides meet again at the Emirates Stadium on Wednesday night, but Coyle is keeping a keen eye on the midfielder.
Asked if his interest in Wilshere was still there, Coyle said: "Always. I admire good players and that kid is certainly a good player."
He added: "I'm a little bit uncomfortable talking about it because we play Arsenal on Wednesday and I don't think that's the way we should go about it.
"But he's a fantastic young player, a player I believe will eventually play in the Arsenal first team and for England.
"He's one that I've always admired, but I think that conversation is for after Wednesday night when we've played Arsenal, and we'll take it from there."
Another player linked with Bolton is veteran Real Madrid midfielder Guti, whose contract at the Bernabeu is up at the end of the season.
If he were to move to the Reebok Stadium, the 33-year-old Spaniard would be following in the footsteps of veteran stars like Ivan Campo and Fernando Hierro that Sam Allardyce brought to the club from Madrid, but Coyle said he had not made any approach for the player.
"Guti has always been a fantastic player, a top player in world football all his days, and I think the job Sam Allardyce did at this football club was unbelievable and all credit to him," Coyle said.
"I think what you do is you recognise if there is a position that you think can help the football club and there is the right kind of player available then you look to strengthen it and bring in accordingly.
"But I haven't had any conversation about Guti with anybody."
Coyle also confirmed he had not made enquiries about Sheffield United centre-half Kilgallon or Stoke right-back Wilkinson, both of whom he has been linked with.
Coyle said: "I'd be quite comfortable telling you if I'd spoken to anybody about any of those players and I haven't.
"I think we all recognise that because there is a transfer window there will be speculation and if a new manager comes in, you're going to be linked with everyone from Guti to whoever else is thrown up.
"If and when I have news, you will be the first to know about it."
One player who could be about to sign on with Wanderers is United States midfielder Stuart Holden, who has impressed Coyle during a trial with the club.
"Stuart did well since he came in," said Coyle.
"He's picked up a little thigh injury in the last couple of days which has curtailed his movement a little bit.
"But he's done well and I'll have a chat with him today and his representative and see where we go from there."
Arsenal 2/9, Draw 9/2, Bolton 14/1