The ex-Queens Park Rangers goalkeeper has been a free agent since departing from the club in May.
Per The Mirror, manager Sean Dyche sees Begovic as a solid option for his squad.
He wants to bring back the experienced ex-Premier League keeper as a backup.
Begovic would likely be third choice, but would be invaluable to Dyche on the training pitch.
He last played for the Toffees back in 2021, spending two seasons at Goodison Park.