Everton lost against Lyon at Goodison Park in the Europa League on Thursday evening.
Everton boss Ronald Koeman is under serious pressure, with the Toffees in 16th place.
Everton signed Jordan Pickford from Sunderland for £30 million yesterday.
The goalkeeping legend had high praise for the French goalkeeper.
The former goalkeeper believes the striker will only leave Everton for another Premier League club.
Everton goalkeeper Joel Robles became the club's first choice last season.
Hennessey has a year left on his Crystal Palace contract, while the Toffees are down to just one senior 'keeper.
David Moyes has been linked with a return to Everton.
Everton are on the lookout for a new manager following the departure of Roberto Martinez.
Everton are having a disappointing season in the Premier League.
The USA keeper cleaned out Andre Ayew by reacting sluggishly to a John Stones backpass.
Neville Southall does not believe that Simon Mignolet is a Liverpool standard goalkeeper.
Everton legend Neville Southall has backed manager Roberto Martinez to turn the club's fortunes around after a terrible start to the new season.