The Gunners are looking to increase their options at centre-back, but have already had one bid for the 29-year-old rejected.
While Arsenal boss Arsene Wenger could return with an improved offer for the England international, the Toffees boss has said that Jagielka will not push for a move to the Emirates.
"I have had two or three conversations with Jagielka 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."
Jagielka joined Everton from Sheffield United in 2007 for a fee of £4m.