Moyes' 11-year reign at Goodison Park came to an end earlier this month when he was chosen to replace Sir Alex Ferguson at Manchester United.
There have been worries that a number of Everton players will follow Moyes out of the club, but Jagielka insists he will not be one of them.
When asked if he still saw his future at Everton, the England defender replied: "Yes, definitely.
"I was made captain for next season when David Moyes was in charge.
"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 are yet to appoint Moyes' successor, although Roberto Martinez has been strongly linked with the position.
Jagielka joined up with the England squad last night ahead of the Three Lions' friendlies against the Republic of Ireland and Brazil this week.
The 30-year-old insists he will be able to concentrate fully on his England duties despite the uncertainty surrounding his club.
"I will be able to put it to the back of the mind, definitely," Jagielka told a press conference at St George's Park today.
"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."
England face Giovanni Trapattoni's side on Wednesday night at Wembley before taking on Brazil at the famous Maracana Stadium in Rio de Janeiro four days later.
With John Terry and Rio Ferdinand now retired from international football, Jagielka is in a good position to cement his place in the England defence over the next few months.
This week he will compete with Gary Cahill, Phil Jones and Joleon Lescott for a place at centre half.
Injury has kept the former Sheffield United defender out of the England reckoning since last October and he is therefore looking forward to a return to the international arena this week.
"It's never nice to get injured during the season, especially during the international breaks," Jagielka said.
"There aren't many international games per year so you want to feature in as many games as possible.
"Sometimes when you get friendlies they are more like testimonials, but I am pretty sure that between England and Ireland it won't be the case.
"(Wednesday's game) is probably going to be more like an English Premier League style game.
"We have a couple of Irish lads in our dressing room so there's been a bit of banter flying around between us."
Danny Welbeck and Kyle Walker missed training this morning, although both players took part in a gym session, raising hopes they could take part in either of the two games.
It is understood Walker was hurt during Tottenham's end-of-season friendly against Jamaica in the Bahamas last Thursday.