Theresa May will not force her ministers to abstain in any vote of no confidence against the Speaker
Mr Bercow angered Tory MPs when he said he would not allow the US President to make a Parliamentary address following his controversial travel ban on seven Muslim-majority countries.
The Speaker's position became more precarious after it was revealed he voted for Remain in the EU referendum, leading to questions over his impartiality.
A Times investigation also claimed Mr Bercow blocked warnings of police probes into shamed MP Keith Vaz, after receiving tens of thousands of pounds from donors allegedly linked to the Labour politician.
Conservative MP James Duddridge has launched a petition calling for a vote on whether Mr Bercow should be allowed to continue as Speaker, with at least 150 Tories said to be planning to vote against him.
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Theresa May could have told her whips to ask ministers to abstain from the vote, but instead they will be free to back the petition if they so wish, the Telegraph reported.
Although the vote is unlikely to win a majority, should it take place, a group of furious Tory MPs are planning to mount a guerrilla campaign and repeatedly call for Mr Bercow’s resignation in the Commons until he is forced to step down.
Mr Bercow's problems started when he was asked by Labour MP Stephen Doughty if he was aware of the grave concerns some MPs had regarding an address by the US President to politicians in the Palace of Westminster.
The Speaker replied: "We value our relationship with the United States.
"However, as far as this place is concerned I feel very strongly that our opposition to racism and to sexism and our support for equality before the law and an independent judiciary are hugely important considerations in the House of Commons.
"Before the imposition of the migrant ban I would myself have been strongly opposed to an address by President Trump in Westminster Hall.
"After the imposition of the migrant ban by President Trump I am even more strongly opposed to an address by President Trump in Westminster Hall."
His speech won applause from opposition MPs, with Dennis Skinner telling the speaker: "Two words: Well done."
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Prime Minister Theresa May will not protect John Bercow form a no confidence vote
John Bercow angered Conservative MPs when he spoke out as Speaker against Donald Trump
Labour has fervently backed the embattled speaker, with deputy leader Tom Eatson calling him "one of the greatest speakers the House of Commons has seen".
Veteran Labour MP David Winnick also defended the Speaker, saying he was a "champion of the backbenchers" who is usually willing to call every MP who wants to ask a question during ministerial statements, and to grant them urgent questions to the Government.
And he predicted the plot, which he claims was "orchestrated by some on the Tory side who just never accepted that he should be in the chair in the first place", would fail.
Even Ukip MP Douglas Carswell, whose motion of no confidence led to the downfall of Michael Martin as speaker in 2009, said the move against Mr Bercow was "misguided".
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