England were woeful last night in their 2-1 loss to Iceland.
The pair worked together extensively at Swansea City.
Everton are looking to sign a new goalkeeper this summer.
Newcastle improved under Benitez towards the end of last season but will start next season in the Championship.
The Paraguayan has been told by his former club they will not be signing him, as Spurs loom.
The 30-year-old scored the Azzurri's second goal to knock the holders out of the competition.
Everton have plenty of money to spend this summer, following the arrival of Farhad Moshiri.
England crashed out of Euro 2016 thanks to a 2-1 loss to Iceland in Nice.
Reports suggest Fulham joined Newcastle United in pursuit of the Crystal Palace hitman.
Crystal Palace are interested in signing Andros Townsend from Newcastle United.