The 50-year-old Scot will officially take charge of Manchester United on July 1 following an 11-year stint at Goodison Park.
After confirmation of Phil Neville's exit at Everton, Moyes had appointed Jagielka as the club's new captain, and the 30-year-old wants to continue the role under the next manager.
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"We will have to wait and see what happens with the new manager, but hopefully he will keep me in that role and it is something that I am looking forward to doing."
Everton have yet to appoint David Moyes' successor, but Wigan Athletic manager Roberto Martinez is the club's first-choice candidate to replace the Scot.
Jagielka, who is currently on international duty with England, insists he is focussed on his role with Roy Hodgson's side and hopes the managerial situation at Everton will be resolved before the start of pre-season training.
"I will be able to put it to the back of the mind, definitely," he said.
"We all knew the situation. It wasn't sprung upon us because we had a couple of games to deal with it towards the end of the season when we knew the manager was leaving.
"I thought as a team and as a club we sorted it really well and gave the manager a great send off. We got over it in the course of those few weeks and I am away with England now so I will look forward to these two games.
"I'm not really thinking about [Everton] until we report back in July and hopefully by then we will have a manager sorted and we will see where it takes us from there."