The Toffees held advanced talks with Nuno Espirito Santo last week, but have so far been unable to conclude a deal.
Owner Farhad Moshiri is open the idea of bringing former Liverpool boss Benitez back to Merseyside despite a potential backlash from supporters.
Everton were not expecting to be in the manager market this summer, envisaging Carlo Ancelotti to be in for the long haul until at least their new stadium was completed in 2024.
But then Ancelotti's former side Real Madrid came calling and the Italian made a swift return to the Bernabeu to replace Zinedine Zidane.
The Spaniard is revered by the red half of Merseyside having guided the club to an incredible Champions League win in 2005. He also won the FA Cup with Liverpool.
Since leaving Anfield in 2010, Benitez has gone on to manage Inter Milan, Chelsea, Napoli, Real Madrid and Newcastle.
His latest managerial role saw him take charge of Chinese side Dalian Professional for two years.