The Telegraph says the 32-year-old is a free agent after leaving Burnley this summer and is on the hunt for a new team.
Mee spent 11 years at Turf Moor and had been skipper at the club for the last three years.
But after his deal expired Mee has been in discussions with Brentford.
With less than three weeks to go until the new campaign Mee is set to make his decision shortly, with Brentford kicking off their campaign away at Leicester on August 7.
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