Moyes has already been heavily backed in the off season transfer market, spending around £55M to permanently sign loan stars Merlin Rohl and Tyrique George and bring in Hayden Hackney from Middlesbrough.
However, Moyes is also looking at least two new full-backs, following the exit of former club captain Seamus Coleman and uncertainty over Nathan Patterson's long-term future on Merseyside.
Patterson's place is unlikely to change in Moyes' plans and the Scotland international could be moved on in the coming weeks.
Previous reports from Sky Sports claimed Tottenham's Djed Spence has been discussed as a target, with a bid potentially tabled after the 2026 FIFA World Cup, following England's elimination in the semi-finals.
Everton could face competition from Serie A champions Inter Milan, and Manchester City Rico Lewis is now being lined up as an alternative, with Enzo Maresca reportedly open to letting him leave for the right price.
The England international has endured injury issues across the last 12 months and his City starting place has been taken by Matheus Nunes at the Etihad Stadium.
Lewis is under contract in Manchester until 2030, so City are under no instant pressure to sell the 21-year-old defender, but Everton are hoping to gain an edge on other interested EPL teams with an initial £30M bid.