The Merseyside club had agreed a fee and concluded discussions on personal terms before the weekend with the 20-year-old defender.
But Sheffield United then delayed the deal when Tottenham matched Everton's £6m bid.
Naughton held talks with Spurs manager Harry Redknapp on Tuesday in Exeter, where he was preparing his team for a friendly.
Aston Villa are also interested, but Naughton has privately expressed a desire to move to Goodison Park, where he would be reunited with former Blades team-mate Phil Jagielka.
Now Moyes hopes to be able to tie up any further details with the England Under-21 defender.
The added complication of another Blades youngster, Kyle Walker, being included in the deal has yet to be finalised.
Sheffield United wanted £8m for the pair, and Walker is believed to have been swayed following discussions with Redknapp. Everton are prepared to sign the youngster and then loan him back to the Yorkshire club.
Moyes' squad was boosted today by the return of Tim Cahill and Joleon Lescott to training, the pair having been given extra time off following their international duties this summer.
Striker Ayegbeni Yakubu has seen a specialist this week to assess his recovery from an Achilles injury. He is now expected to step up his rehabilitation but is unlikely to be fit for the start of the season.
Mikel Arteta, Victor Anichebe and Jagielka are also progressing well following knee injuries that ruled them out of the end of last term.
Everton's next friendly is at Rochdale on Saturday, before they fly to Canada and the USA for further warm-up matches next week.