Ministers are now being questioned in the Commons about Monday's critical report by Dame Laura Cox which said harassment allegations had not been dealt with properly because of a culture of "acquiescence and silence".
Mr Bercow has also faced personal allegations of bullying, which he strongly denies.
The BBC and other British media outlets reported that Bercow had told friends of his plan to quit next year.
Conservative MP Maria Miller told Mr Bercow a "change in leadership" was needed at the top "including you".
The remain-supporting speaker has repeatedly attracted controversy during his time in office, hitting the headlines in 2017 for calling a prominent Brexiteer - and now leader of the House of Commons - Andrea Leadsom a "stupid woman".
Former Tory minister Mrs Miller said the report showed that "bullying and harassment is coming right from the top" and it is not right for Mr Bercow to oversee reform.
He has since said he means to stay on and see Parliament through the process of Brexit, which is due to conclude with the UK's exit on 29 March 2019.
Tory James Duddridge, a critic of Mr Bercow, mocked him, listing the "unacceptable behaviours" set out in the report before asking: "How can we encourage Mr Speaker to stop this behaviour?"
Mr Bercow's three predecessors, Bernard Weatherill, Betty Boothroyd and Michael Martin, each spent between eight and nine years in the role.
Speaker John Bercow has called for an independent and external body to judge upon allegations of bullying and harassment in the wake of a report, which found "urgent and serious problems" in the way abusive behaviour by MPs and staff is dealt with.
Although Dame Laura did not investigate individual cases, she said highly-placed figures should consider whether they could change the culture in Parliament and, if not, "consider their position".
"We have outstanding allegations directly against the Speaker who will be one of the first people who will be considering this report". "They wanted to use the information privately to help control their MPs".
Several MPs have warned that the issue of inappropriate behaviour in the Commons must not be politicised in partisan wrangling over Mr Bercow's own future. "Nothing fills the victims with more dread than when people play with their feelings so don't do it".