Chelsea centre-back Terry announced his international retirement on Sunday night on the eve of his Football Association disciplinary hearing on a racism charge claiming the case made his position with England "untenable" - a suggestion the FA have denied.
Jagielka has had to bide his time in the squad behind the likes of Terry, Rio Ferdinand and Joleon Lescott but now has the chance to become a regular starter under Roy Hodgson.
"Without doubt it is a chance for Phil to stake a claim, it is a terrific opportunity, it really is," said Round.
"He has speed, awareness, and is a very good defender one versus one - which is very important in international football.
"He is also starting to show the composure now which has taken him a while to come to terms with: being able to become really comfortable on the ball, taking the ball from the goalkeeper and building from the back.
"He was starting to show that for Everton and he was starting to show it for England.
"There will be one or two thinking they have an opportunity, like (Gary) Cahill at Chelsea.
"It is great opportunity for Jags and if he gets it he will take it.
"He is ready for it - I know that - and he feels he can step up that level.
"He certainly has the talent and ability, he just needs to get in now and establish himself."
Terry's decision to quit England came as a shock to Round, who has previously worked with the defender in the international set-up.
"I was surprised, like everyone else. I worked with him at England for a couple of years and I found him a terrific captain, whole-hearted and 100% every day, every game," said the Toffees assistant boss.
"I couldn't fault him on any issue when I worked with him personally.
"Obviously he feels it is something he has to do but I can't comment on the reasons why he has done it.
"He will be missed because he's a top player and a top defender."
Jagielka comes up against one of his former managers in Neil Warnock on Tuesday when Everton travel to Elland Road in the third round of the Capital One Cup.
The pair were together at Sheffield United and the centre-back remembers those times fondly.
"He is the sort of character you either love or hate but he was the man who gave me my first big chance," he told evertontv.
"He will always have a special place for me as you don't forget those people who help you in your career.
"You want characters in the game, you don't want to be watching interviews on the telly and switching them off because they are boring and Neil Warnock is certainly not boring."