According to BBC Sport, Rodriguez has flown to the Middle East to negotiate a permanent move to a club in Qatar, where the transfer window is open until September 30.
Despite only joining Everton last summer, the Toffees are prepared to let the 30-year-old leave as they are keen to remove his £200,000-a-week salary off their wage bill.
Rodriguez, who was linked with a return to Porto over the summer, is yet to feature under Benitez this season, and has seemingly played his final game for the club.
The former Real Madrid and Bayern Munich midfielder has made 26 appearances across all competitions for Everton, scoring six goals and providing nine assists. body check tags ::