The Basle defender has been tracked by Arsenal boss Arsene Wenger, and impressed watching scouts during Basle's Europa League quarter-final triumph over Tottenham earlier this month.
But the report claims Dragovic is ready to snub a move to Arsenal in favour of Everton. Borussia Dortmund are another who've been linked with the Austrian, but he's thought to prefer a switch to Goodison Park.
David Moyes is hoping to add to his defensive group this summer, with club captain Phil Neville set to depart and uncertainty hanging over the future of Johnny Heitinga.
But Arsenal are also thought to be in the market for a new centre-back. Thomas Vermaelen's dip in form has seen him relegated to the bench, with Per Mertesacker and Laurent Koscielny becoming the club's first-choice partnership under Wenger in recent months.
The former-Austria Vienna defender could be available for £4m at the end of the season, putting him well within Everton's transfer budget.
The 22-year-old is set to face another tough test in the Europa League when Basle take on Chelsea in the first-leg of their semi-final in Switzerland.
Dragovic will come up against either Fernando Torres or Demba Ba, and watching scouts will likely be taking great interest in how he gets on.