Everton are looking for a new manager, following the dismissal of Ronald Koeman earlier this week.
Allardyce has actually been linked with the job at Goodison Park, but the former West Ham and England boss has recommended that the Toffees go for Burnley boss Dyche.
“It goes back to what happens last time. Mosey went from Preston, when they were in the Championship,” Allardyce said.
“Sean has done fantastically well at Burnley for five years now - so let’s see what happens.
“Mr. Kenwright is still there, and maybe he might look to employ a British manager. He’d certainly fit the bill, certainly.”
Dyche was asked about whether he would take the job at Everton yesterday, but insisted that he was focused on managing at Burnley.
The 46-year-old boss has done a remarkable job at Turf Moor, with his ability to organised a side earning him a lot of plaudits.
Burnley actually beat Everton earlier this season, and their display highlighted Dyche’s managerial qualities.
At the moment, David Unsworth is the bookies’ favourite to land the Everton job on a permanent basis.
Unsworth is currently in caretaker charge and will lead the Toffees against Leicester City on Sunday.