BBC Parliamentary correspondent Mark D'Arcy said other senior backbenchers doubt that Mr Duddridge's tactics would work, and said that Mr Bercow has a reservoir of support on the backbenches - particularly among pro-Brexit Conservative backbenchers - and was seen to have enabled MPs to challenge the government far more effectively.
Fellow Tory MP Alec Shelbrooke backed the motion, saying that although Mr Trump's faith-based migrant ban is discriminatory and wrong, the Speaker had "politicised the office and his position is untenable".
Criticising Mr Bercow's conduct as a whole in the chair, he added: "This has been happening more and more often from this modernising Speaker".
More liberal-minded MPs, however, defended Bercow's remarks, noting how offended they were by some of Trump's rhetoric.
The Lord Speaker, Lord Fowler, one of the three key people who would have to approve a speech in parliament, also criticised Speaker John Bercow for not consulting him.
In addition to labeling Mr. Trump as racist and sexist, Mr. Bercow took aim at the president's recent executive order that briefly imposed travel restrictions on citizens of seven predominately Muslim countries.
Mr Duddridge, Conservative MP for Rochford and Southend East, had already written to the prime minister, asking for MPs to be given a free vote if there was a vote of no confidence in the Speaker.
Parliament began a 10-day recess Thursday.
But it now appears that Mr Bercow's stance has been accepted as the Government has reportedly abandoned the plans. Duddridge said there are a number of possible ways that could happen. "Well done, Mr Speaker", said the SNP's Alex Salmond.
We value our relationship with the United States.
"I just have no idea why he said it, especially when at this stage we have no idea when Donald Trump will be coming here, an invitation hasn't even been sent at this stage".
Earlier this week Bercow told MPs that Trump should not be allowed to address Parliament because the USA president is "racist" and "sexist".
Bercow said on Monday: "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".
"I do not intend to argue the case for or against Mr. Trump's visit - that is not my role as Speaker".