The 28-year-old has spent the last season on loan at the French club having been deemed surplus to requirements by Liverpool manager Rafael Benitez.
However, Bordeaux have chosen not to take up their agreed option of buying the player on a permanent deal worth around £2million.
Cheyrou, who still has a year left on his contract at Anfield, has no future at Liverpool and must now find another club.
He was bought for £4.5m from Lille by then manager Gerard Houllier in the summer of 2002, but struggled to make an impression and has not made a single competitive appearance under Benitez, who shipped the Frenchman out on loan to Marseille for the 2004-05 season.
------------------------------------------------------------
Great, will we ever get rid of the sh!te Houllier left behind
