Former Pool boss Clark joins Bury
Clark, who resigned as Blackpool boss in May 2015 following the club's relegation to League One, had been in charge at Kilmarnock since February 2016.
He kept Killie in the Premiership last season and leaves them in sixth place.
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Hide AdBury, meanwhile, are in the relegation places in League One, two points from safety.
Clark only managed to oversee three Blackpool wins during his 33-game spell at Bloomfield Road after replacing Jose Riga in a season punctuated by regular fan protests against the running of the club.
The 44-year-old has previously managed Birmingham City and Huddersfield Town