The Bills announced the moves Friday in placing Bush on the reserve/retired list. Bush had seven starts in 15 games during his only season with Buffalo last year. He had one year left on the two-year contract he signed with the Bills in March 2018.
Bush, 32, spent five seasons in New Orleans and had 25 starts in 91 career games.
Coleman, 31, has nine seasons of NFL experience and had nine starts in 16 games with New Orleans last year. He also spent three seasons with Carolina, where his tenure overlapped with Bills coach Sean McDermott, who served as Panthers defensive coordinator.
Buffalo opens training camp in suburban Rochester, New York, on Thursday.