The motion has been put forward by Tory backbencher James Duddridge, who responded furiously to Mr Bercow's comments on Monday.
He said the Speaker's comments were just the latest in which he ignored his position's expectation of political neutrality.
Mr Duddridge "He has overstepped the mark, he has overstepped the mark a number of times but this most recent incident – where he used the Speaker's chair to pronounce his views on an international situation in some quite detailed and lengthy manner is wholly inappropriate and it means that he can no longer reasonably chair, as Speaker, any debate on those subjects.
"This has been happening more and more often from this modernising Speaker.
"This is perhaps the straw that has broken the camel's back."
The motion was tabled as the Commons rose for the February recess, with Mr Duddridge claiming support for the move could see the Speaker forced out before MPs return to Westminster.
John Bercow is to face a motion of no confidence following his Donald Trump comments this week
Despite holding the position of Speaker of the House of Commons, a role which demands objectivity, Mr Bercow said on Monday he would not invite Donald Trump to speak to MPs due to allegations of racism and sexism.
In a speech lasting several minutes he lambasted the US president before finally re-taking his seat, receiving rare applause from opposition MPs.
This provoked a furious response among Conservatives including Mr Duddridge, whose motion heaped more pressure on the Speaker.
Mr Bercow's comments were defended by Labour today, who said his comments were valid.
Shadow Commons leader Valerie Vaz said in Parliament today: "When a person refers to a senator, Elizabeth Warren, as Pocahontas, who is then silenced by her party. When a person repeats the cry, 'Lock her up', of a candidate, when no offence has been committed.
John Bercow's best quotes
Wed, February 8, 2017
Outspoken John Bercow, the Speaker of the Commons, has both wise and not so wise words.
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The Conservatives must realise that being sceptical is different from being phobic in what is an interdependent world.
"When a person suggests women should be grabbed in certain places without their consent. When a person has consistently questioned the birthplace of an American president, President Obama."
She went on: "Then I, born in Aden, Yemen, of Goan-Indian heritage, who may or not be directly affected by the travel ban, and others, welcome the support for us and for the reputation of Parliament.
"Could the leader of the House confirm that the Government will not support any attempts to act on the letter to the Prime Minister about comments made on a point of order in this chamber?"
John Bercow has been criticised for his comments on Donald Trump
Commons Leader David Lidington did not respond to the question about the Speaker but said the Government had to deal with the US president as he was democratically elected, despite strong feelings on the matter.
He said: "Whatever view any of us as individuals might have on any particular leader of another country, the reality is that governments have to deal with other governments in the world as they exist and particularly with elected governments who are able to claim a mandate from their own people."