Beardsley has left Newcastle United 14 months after taking a period of leave so the club could launch an investigation into his conduct.
The club, in a short statement, thanked Beardsley – who had been coaching the Under-23s – for his "contribution as a player, coach and ambassador".
Beardsley, speaking through his solicitors, said he was looking for a "new challenge".
However, neither statement addressed the allegations of bullying and racism made against Beardsley by a number of players, notably Yasin Ben El-Mhanni.
Beardsley, 58, categorically denied the claims, which are still "under consideration" by the FA.
An FA spokesperson said: "The allegations made against Peter Beardsley continue to be under consideration by the FA. We are not in a position to comment further at this time."
Beardsley reportedly agreed a financial settlement before leaving United.