If they don't he will quit October 31 - the day Britain is due to leave the European Union.
A lifelong Conservative, he angered many in his own party - particularly in his handling of parliamentary debates around Brexit - to the extent Tory grandees announced they would field a rival candidate in the next general election in a bid to unseat him.
"He has been dogged by controversy about his alleged bullying behaviour, he has undermined faith in politics and there is no intention on our side to nominate him for a peerage", the source said.
Addressing the Commons while his wife looked on from the gallery, Mr Bercow said: "At the 2017 election, I promised my wife and children that it would be my last".More news: Iran's Full and Timely Cooperation is Essential, Acting Director General Tells Board
Mr Bridgen continued: "This is a zombie Parliament, full of zombie politicians, including a zombie Speaker". This is a pledge that I intend to keep'.
If the House does not vote for an early election, Bercow's last week on the job will likely be a tumultuous one.
The Labour MP and Mother of the House - the longest continuously-serving female MP - made the announcement after the current Speaker, John Bercow, said he would stand down by 31 October.
Opposition MPs gave Mr Bercow - who will walk away with a gold-plated £1million non-contributory pension after ten years in the chair - a standing ovation. "Our democracy is the stronger for your being the Speaker", Mr Corbyn said.More news: Zimbabwe delegation holds mass for Mugabe in Singapoore
While Mr Bercow moves on to a new challenge, Coral has also offered the latest odds on the next Commons Speaker, with a Labour replacement touted.
The Fixed-term Parliaments Act 2011 introduced fixed-term elections to the Westminster parliament for the first time, as a result of the Conservative-Liberal Democrat coalition agreement which was produced after the 2010 general election.
He added it would take place regardless of the outcome of a government-led vote on holding a snap election next month.
The new legislation, which received royal ascent on Monday, changes that, requiring the Prime Minister to ask the European Union to postpone the exit until January 31, 2020, unless parliament approves an agreement, or exits without agreement by October 19.More news: King’s Korner: Oakland Raiders start the season with a statement win
"Parliament will be prorogued at close of business today", a spokesman for Prime minister Boris Johnson said, using the parliamentary term for the suspension.
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