The statement comes after his former Clerk of the House, Lord Lisvane, made a formal complaint about him to the Parliamentary Commissioner for Standards.
But Mr Bercow said he had never "bullied anyone, anywhere at any time".
The former Speaker also criticised the government for failing to nominate him for a peerage.
He said that, since retiring, it had "become increasingly obvious that the government has no intention of honouring the centuries-old convention that a departing Speaker is promptly elevated to the House of Lords".
Mr Bercow added: "Indeed, it has been suggested to me that the government actively seeks to block any other attempt to nominate me for membership of the upper House."
There have been reports that the former Speaker - who previously sat as a Conservative MP - has been nominated for a peerage by the outgoing Labour leader Jeremy Corbyn.
But Downing Street said appointments to the House of Lords are "a matter for house authorities" and they were "not aware of a nomination being put forward".
A No 10 source suggested to the BBC that Mr Bercow's accusation showed a lack of understanding of the process of nominations for a peerage, adding: "The speaker wasn't always a fan of convention."
Lord Lisvane, who served as Clerk of the House between 2011 and 2014, confirmed to the BBC he had submitted a formal complaint on Wednesday, which was understood to be centred around bullying.
In an earlier statement, Mr Bercow - who stood down in 2019 after 10 years in the chair - said his former colleague had "ample opportunity" to raise issues in their time working together, adding that the timing of the intervention was "curious".
On Thursday afternoon, he said: "I have seen in the media that Lord Lisvane is formally complaining that I bullied staff.
"For the record, I categorically deny that I have ever bullied anyone, anywhere at any time."
Mr Bercow has faced other accusations of bullying during his time in office, but has denied all the claims.