Garner, 24, has been one of the Toffees' most consistent performers, playing every minute of the Premier League campaign so far. His versatility has been a major asset, with Moyes using him both in central midfield and at right-back.
The former Manchester United academy graduate helped Everton to a 2-0 win over Fulham last weekend, continuing a run of strong displays that have caught the attention of many observers.
Now in his fourth season at Goodison Park, Garner has represented England at every youth level up to the Under-21s, where he played a key role in Lee Carsley's European Championship-winning side in 2023. A senior call-up has yet to arrive, but Moyes believes his time could come soon.
"He's been fantastic since I came in," said Moyes. "I probably didn't realise how good he was until I started working with him. He's played great in midfield and done a brilliant job at full-back too. With the way he's performing, he's definitely getting closer to being talked about for England.
"England have so many quality players to choose from, you only have to look at Foden and Bellingham, but Jimmy is in that group of young midfielders who could get a call-up in the future."
Garner himself has made no secret of his ambitions, saying recently: "I want to get into that England team, that's my long-term goal. I know it won't happen overnight, but I definitely feel like I've got the quality."
Moyes also praised Jack Grealish, who is on loan and pushing to regain his England place after more than a year out of the squad. The winger has made a major creative impact, recording four assists and 24 chances created, more than any other player in the Premier League.
"Jack has been really good for us," said Moyes. "His assists have been huge, but I'm pushing him to add more goals to his game. He's helped us keep the ball better and evolve our style.
"Thomas Tuchel has a tough job picking his England squad, but Jack wouldn't let anyone down if he's called up."
