Sir Keir Starmer is facing a possible parliamentary investigation over allegations he put pressure on the Speaker in a debate on Gaza last week.

Sir Lindsay Hoyle is facing a backlash for allowing a vote on a Labour amendment to an SNP motion calling for an immediate ceasefire in Gaza.

Parliamentary convention dictates that there would usually only be a government amendment to an opposition motion, but Sir Lindsay said he selected the Labour amendment to allow as broad a debate as possible.

However, critics within the SNP and the Conservatives have claimed he bowed to pressure from the Labour Party to select the amendment with the aim of staving off a potential rebellion among its MPs who could have voted for the SNP motion if denied the opportunity to vote on their own.

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Following the outcry, reports circulated that Sir Keir had put pressure on Sir Lindsay, a Labour MP before taking on the Speaker role, to select his party’s amendment in order to stave off a potential rebellion – thus bringing his impartiality into question.

While Sir Keir has “categorically” denied the claims, Sky News has learned that the Commons leader, Penny Mordaunt, believes there could have been a “breach of privilege” and an investigation is one of a number of potential options being considered.

Asked on Monday if he regretted the way things had panned out, the Labour leader said: “My focus is on the awful situation in Gaza. Not the parliamentary process, the awful situation.

“And we all want to see an end to the thousands of people being killed in Gaza. We want to see those hostages out, and we want a pathway to a peaceful settlement.”

Sir Lindsay has also rejected accusations he was put under pressure by Labour and has insisted the safety of MPs was the main reason for his move. He later issued an emotional apology admitting he had made a “mistake”.

On the prospect of a privileges committee probe – first reported by the Times – a Labour spokesperson said it was “desperate stuff from a Tory party trying to distract from their own troubles by repeating lies about Keir Starmer”.

Sir Lindsay is facing a battle to save his job following the debacle, which has led to the SNP – the third largest party in the Commons – losing confidence in him.

A total of 81 SNP and Conservative MPs have now signed a petition of no confidence in Sir Lindsay.

The SNP’s anger was stoked further when the Speaker rejected an application from the SNP for an emergency debate over a ceasefire in Gaza – something Sir Lindsay himself had proposed as an olive branch following the scenes last week.

Sir Lindsay said the government planned to “make a relevant statement” around the situation in Gaza on Tuesday, meaning there would be a “very relevant opportunity for this matter to come before the House”.

But the SNP’s Westminster leader, Stephen Flynn, accused parliament of “failing the people of Gaza by blocking a vote on the urgent actions the UK government must take to help make an immediate ceasefire happen”.

“The Speaker broke the rules last week – and this week he has broken his word,” he said.

“How can MPs have any trust in the Speaker when he makes a public commitment one minute, only to rip it up the next?

“If 30,000 dead Palestinians aren’t worthy of an emergency debate, what is?”

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Labour’s role in last week’s saga came back into focus this week following an interview shadow minister Chris Bryant gave on Channel 4 News, in which he admitted to filibustering – a delaying tactic – ahead of the opposition day debate to allow Sir Keir and the Speaker time to talk.

The SNP’s Kirsty Blackman said Starmer’s party had been “caught red-handed following the admission by Chris Bryant”.

“There must now be a full, independent investigation into the appalling behaviour of Keir Starmer and his colleagues, who are no better than the Tories when it comes to manipulating the broken Westminster system,” she said.