The 18-year-old joined the Trotters from the Gunners in January and after impressing in his 14 games Coyle is determined to secure his services once more.
Coyle told the Bolton News: "I called Arsene Wenger to thank him for letting us have Jack last season. I think it was a fantastic experience for him and a very worthwhile one for us too.
"I honestly think he improved as a player, so we have asked the question. If he feels that Jack is not quite ready for first-team football [at Arsenal] next season, then we would love the opportunity to bring him back.
"The fans can see what a great young English talent he is, and I believe that with another full season under his belt, that he can really push on.
"That is how we left it - so we'll both go away and take a break, and then see what happens."
Coyle also revealed that Gary Cahill will not be used as bait to extend Wilshere's loan spell from Emirates Stadium.
"There has been no dialogue about Gary Cahill whatsoever, with Arsenal or anyone else for that matter.
"I have got no desire to lose Gary - he's a big, big part of what I am trying to do here."
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