Jagielka impressed in Sunday's Merseyside derby draw with Liverpool and produced an outstanding showing in front of England coach Ray Lewington.
England boss Roy Hodgson has been struggling to find a first-choice pairing at centre-back since John Terry retired and Rio Ferdinand dropped out of favour, but Moyes believes Jagielka can be the answer to Hodgson's problems.
"When you look at the situation with England at the moment, it's there for Jags to go and say 'pick me'," said Moyes.
"He was top drawer at Anfield and has been like that for a long time.
"He sometimes doesn't believe in himself enough or have the confidence he should have when you see how he plays.
"But, as a manager, he gives you everything.
"He's been absolutely superb for us - he can be a real leader and always gives everything in every game he plays in."
Jagielka captained Everton against Liverpool in place of the injured Phil Neville and will take up the role on a permanent basis next season when Neville leaves Goodison Park.
Moyes believes the former Sheffield United man can develop into a leader for Everton, adding: "He'll get better with responsibility over time, and we hope he'll be a great captain."