However, Coyle insists defender Gary Cahill will not be part of any package after he had been linked with the Gunners.
Coyle said: "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 added in the Bolton News: "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."
Source: Team Talk
Source: Team Talk