Lecce's sporting director Pantaleo Corvino said: "We hope to sell him to an Italian club, not abroad like Everton, because that is what he wants.
"At the moment he will stay at Lecce and hopefully he can repeat the same wonderful season he had last year."
Vugrinec, who moved to Italy from Trabzonspor in Turkey, has played in 13 games this season for Lecce and scored three goals, but currently has a back injury.
He may be being lined up as a replacement for Thomas Gravesen, Everton's unsettled midfielder who has asked to leave the club.