Andrea Leadsom, who as leader of the Commons clashed repeatedly with Mr Bercow, said the complaints procedures for parliamentary staff - which she helped establish - applied to everyone in the House.
Lord Lisvane, who was clerk of the House, has handed a dossier of allegations to the Parliamentary Commissioner for Standards, according to the paper.
Mr Bercow dismissed the claims as having come at a "curious" time, with suggestions that he may be in line for a peerage.
Mrs Leadsom told Sky News: "The whole purpose of that was that anybody, including the Speaker of the House of Commons whoever he or she may be, should also be subject to that procedure.
"In the event that there are genuine, upheld complaints about any persons that should have an impact on whether they are found suitable for the House of Lords."
The document is said to accuse Mr Bercow of having bullied and humiliated staff, including using inappropriate language.
Mr Bercow has consistently denied allegations of bullying from former members of staff in the past.
In a statement on Thursday, he said: "During the five years that we worked together, Lord Lisvane had ample opportunity to raise any accusations of bullying with me.
"At no stage did he do so, even though he became Clerk of the House - the most senior official. The timing of this intervention is curious."
Outgoing Labour leader Jeremy Corbyn has reportedly been lining up Mr Bercow, who was a Tory MP before becoming speaker, for a peerage.
But there have been suggestions that Downing Street could hamper the move, with the Prime Minister's official spokesman highlighting a "long-standing convention" that opposition leaders nominate individuals form their own parties.
Mr Bercow left the speaker's chair on October 31 and has been replaced by Sir Lindsay Hoyle.