Former minister Crispin Blunt, a strong Leave supporter, is seeking support for an early day motion (EDM) when MPs return to Westminster on Tuesday following the Easter break.
EDM's - which are not allocated any time in the parliamentary timetable - are rarely debated or voted on, but are used by MPs as a vehicle to express their views on a particular issue.
Strong support for a motion of no confidence would be highly embarrassing for Mr Bercow, adding to the pressure on him to say when he intends to step down.
Mr Blunt's intervention follows a series of controversial rulings by the Speaker which were widely considered to favour Remain supporters.
Tory Brexiteers were further angered by reports that Mr Bercow - who had previously indicated he would stand down in the summer after 10 years in the post - now intends to carry on.
Mr Blunt said in his email that he hoped to get a "minimum" of 100 MPs to sign to make it a "substantial motion" and to provide "cover" for others who wished to join them.
In a statement, Mr Blunt acknowledged they were in "totally unprecedented territory", but said the impartiality of the Speaker was an "indispensable condition" for workings of the House.
He said: ”Even his most partisan supporters for the positive changes he has delivered as Speaker do not now seriously dispute his bias in the conduct of our affairs.
"But it now goes way beyond the Brexit debate. The public should never underestimate how far reaching the implications of these powers are for our democracy.
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"The holder of this great office decides which MPs can speak in the House of Commons Chamber during debates and selects the amendments that MPs vote on, giving the Speaker the effective power to shape legislation.
"If colleagues are too cowed or too content with the direction of his bias to state their view on Speaker Bercow's obvious partiality, then not only will he feel able to continue, as is being reported in the media, but also feel able to do so exercising the power of his office wholly inappropriately.
"MPs will have failed to stand up for our most basic standards and have no one to blame but ourselves.”
Mr Bercow's office declined to comment on the statement.
Who is John Bercow?
The current Commons Speaker has been in the post for nine years and has been something of a reformer in Parliament.
The 56-year-old former Tory MP probably has more support on Labour benches for his perceived stance on Brexit, putting him in the firing line of Eurosceptics.
However, he has also won praise for making it easier for backbenchers to grill the Government - and consistently allows the 30-minute PMQs session to overrun to at least 45 minutes every week.
Mr Bercow, whose wife Sally was a contestant on Celebrity Big Brother in 2011 and is also a Remain supporter, is perhaps the most famous Speaker of recent times.
As the Brexit chaos made worldwide headlines, the scenes of Mr Bercow controlling rowdy MPs with his own forceful shouts of “order!” have given him a cult following.
As Theresa May faced a vote of confidence in her leadership in January, Mr Bercow’s shouting became a viral sensation and earned him praise across the continent.
German weekly Die Zeit praised “the power of the eighth dwarf”, while Portuguese newspaper Publico said that the Speaker “does not use a hammer, but it seems like he does”.
The Canadian Broadcasting Corporation also published a video to Twitter, titled “British Parliament’s House Speaker is not to be trifled with”, which highlighted some of his most thunderous and pithy pronouncements.
One CBC News editor wrote in response: “Sitting in the newsroom today and hearing that voice, I thought someone in the control room decided they’d rather watch a ‘Pirates Of The Caribbean’ movie than news.”
Nevertheless, the man who has infuriated many a Prime Minister has also had his own fair share of criticism.
In 2018 he faced allegation of bullying after his former private secretary, Angus Sinclair, told BBC Newsnight that Mr Bercow shouted and swore at him and tried to physically intimidate him.
More members of staff have also made similar accusations, all of which Mr Bercow denies.
After being in the post since 2009, Mr Bercow had been expected to resign this year but he has decided to stay in place while the Brexit crisis plays out - unless Tory MPs get their way…