West Ham booked their spot in the fourth round of the FA Cup on Sunday afternoon after beating Leeds United 2-0.
Manuel Lanzini's goal in the first half put the hosts in control, before Jarrod Bowen secured the win deep into stoppage time.
"If he keeps playing at the level he's playing at he's going to get closer," said Moyes.
"The competition is so strong in that position, but if you've got wide players who score goals and make assists he's got a chance.
"Hopefully he'll keep that pressure on Gareth. As a manager you enjoy the thought of helping someone become an international.
"We'll keep pushing Jarrod, keep trying to get him scoring and improve his game, and hopefully get him an England cap."
Bowen has been superb for the Hammers this season, with Sunday's goal marking his sixth of the campaign, while he's also managed 10 assists in his 29 appearances.
He came close to adding further goals to his tally, with a shot cleared off the line in the first half, before Illan Meslier superbly stopped his strike in the second.
The goal his display deserved eventually came in the 93rd minute, when Leeds United piled players forward for a corner, only for Michail Antonio to lead a punishing counter-attack which ended up at Bowen's feet.
He was also heavily involved with the opener, interfering from an offside position before agonizingly waiting for VAR to confirm that he hadn't actually touched the ball.
It was a moment of controversy that clouded the game, but Leeds United head coach Marcelo Bielsa quickly distanced himself from the drama, "It's a decision I prefer to accept what the referees say.
"Not because I haven't created an opinion, but it's a collaboration us managers should make, (we) should accept their decisions even if they jeopardise us."
After booking their spot in the next round, West Ham didn't have to wait long for their next opponents to be revealed.
However, the Hammers managed to avoid Premier League opposition and will instead travel to non-league Kidderminster.