Last week the Toffees agreed a fee with the Yorkshire club and the 20-year-old was given permission to talk to the Merseysiders over personal terms.
Naughton played for the Blades on Friday in a pre-season friendly in Malta, with manager Kevin Blackwell saying: "Sadly, that was his last game for the club. I don't want to lose my best players but the clubs have agreed on a fee."
But since then Tottenham have made a late bid for the player, with Aston Villa apparently also waiting in the wings.
Everton were expecting Naughton at Goodison Park on Monday for a medical, but it is now believed that his advisors are also discussing terms with Tottenham as well as the FA Cup finalists.
The Toffees believe that Naughton wants to join them, and are far from happy with the way the deal has been derailed over the past few days.
Everton had agreed a £5million fee, but Spurs are believed to have improved on that, and the negotiations are now expected to drag on for a few days yet.
Everton have also been linked with Reading winger Stephen Hunt and Cardiff midfielder Joe Ledley as boss David Moyes bids to boost a squad that finished fifth in the Premier League last season.