Jagielka has already been the subject of one from Arsenal this summer, and it has been suggested the Gunners could return with an improved offer before the end of the month.
Moyes, though, has been assured by Jagielka that he will not be pushing for a move.
"I have had two or three conversations with Jags and he is up-to-date with everything and anything that has happened," Moyes told the Liverpool Echo.
"He has said to me that, unless it is something Everton want to do, he is not going to be the one who instigates or pushes anything."
"It will only be if Everton want to do it and I certainly don't want to do it."
Meanwhile, Moyes has revealed the club are close to agreeing a new contract with midfielder Marouane Fellaini.
Everton have been locked in talks with the Belgian all summer about a contract extension and Moyes is hopeful the deal will be concluded soon.
"I think the chairman is moving much closer to agreeing a deal with Fellaini," added Moyes.
"I think we are moving towards that now. I can't tell you exactly whether it will happen but there have been several conversations and I think we are moving much closer towards that now."