Albrighton comes off contract at Villa next month.
The 24-year-old wideman is admired by Roberto Martinez and the Toffees boss is ready to offer him a fresh start at Goodison Park, says the Birmingham Mail.
Albrighton is keen to wait until the uncertainty surrounding the club's future is resolved before committing himself to a deal.
There are also believed to be a string of Championship clubs monitoring Albrighton's situation.
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