Moyes has concerns about their lack of goalscoring form this season and is eager to bring in a new centre-forward in January.
Sky Sports' Rob Dorsett is reporting Manchester United striker Josh Zirkzee and Arsenal's Gabriel Jesus are both on Everton's radar.
A loan move for Zirkzee could be on, though Everton face competition from West Ham and Serie A clubs for the former Bologna star.
Meanwhile, Gabriel Jesus is also interesting Everton, though much will depend on whether Arsenal boss Mikel Arteta can be persuaded to let the Brazil international leave.
Gabriel Jesus returned to training last week after suffering a knee injury last season.