The 34-year-old had most recently been playing in the Singapore top flight with Tampines Rovers.
Pennant began his career at Notts County and has played for 13 different clubs, including Leeds and Wigan.
He has been training with the League One Shakers for several weeks, but they still need international clearance before he can make his debut.
Pennant was signed by Arsenal from Notts County for £2m just before his 16th birthday in January 1999 - at the time a record fee for a trainee.
His most recent appearance in English football was for Wigan during the 2014-15 season before moving to India and then Singapore.
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Bury have signed ex-Liverpool, Arsenal and Stoke winger Jermaine Pennant on a deal until the end of the season.