Free agent Mee, 32, left the Clarets in June when his contract expired and has been targeted by the Bees ever since.
Sky Sports says manager Thomas Frank's side have reached a deal with the centre-back, who is set to become their fifth signing of the summer.
The versatile defender, who can also play at left-back, is set to join Aaron Hickey, Keane Lewis-Potter, Yehor Yarmoliuk and Thomas Strakosha.
The 32-year-old has reportedly turned down offers from MLS as well as two other Premier League clubs.
Frank is currently short of bodies at the back following injuries to key men Ethan Pinock and Kristoffer Ajer.
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