Labour is considering appointing an independent reviewer to oversee its handling of antisemitism complaints as the party battles to stem the flow of resignations after a ninth MP announced his departure on Friday.

Lord Falconer said he was in advanced conversations with the party’s general secretary, Jennie Formby, about taking up a new position in which he would examine whether the party was handling antisemitism and other complaints fairly. The former lord chancellor, who is respected on all wings of the party, said he had approached Formby with his proposal “because of my horror that the party has not been able to show it is dealing with antisemitism properly”.

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Falconer’s potential appointment emerged hours after Ian Austin, who represents Dudley North, said he had resigned as a Labour MP, complaining that a “culture of extremism, antisemitism and intolerance” had taken over the party.

Austin said he had no plans to join the newly formed Independent Group, of eight ex-Labour MPs and three former Tories, although his decision to quit was welcomed by several of its members as a brave and difficult one.

The outgoing MP, who has long been a critic of Jeremy Corbyn, said he did not want the Labour leader to enter Downing Street: “I always tell them the truth and I could never ask local people to make Jeremy Corbyn prime minister.”

Corbyn denied on Friday night that there was widespread bullying in Labour. “There is no place for harshness, bullying or anything else in the party. I don’t believe that it exists on a wide scale,” he told Sky News. “Where there is bad behaviour, we deal with it. Where there is a problem we deal with it.”

Labour called on Austin to resign his marginal seat and trigger a byelection, but he said he would not do so. An MP since 2005, Austin hung on to his seat in 2017 by just 22 votes over the Conservatives.

Party insiders had thought that Austin was the most likely to quit of Labour’s remaining disaffected MPs, but there is intense concern in some quarters that Corbyn needs to engage with the issues raised by the defectors or risk more resignations in the coming days.

Labour sources said Tom Watson, the party’s deputy leader, was speaking to MPs who had been critical of Corbyn in the past to establish what changes they would like to see to halt any further defections.

One idea being pushed is a reshuffle of the shadow cabinet to broaden its base within the party, although Corbyn has not been interested in making changes to the senior team following the relatively successful 2017 election campaign.

Chris Leslie, one of the seven MPs who quit Labour on Monday to join TIG, says his former party had been “hijacked” by a clique. The Nottingham East MP told the Guardian he believed Corbyn was promulgating “hard-left” policies that did not deserve the label of “Labour”.

“There’s a very well-organised clique of ideologically determined people inside the Labour party who have captured the leadership, and the national executive, and obviously tried to encourage those who weren’t in the Labour party before 2015 to come in and take over,” he said.

The attack by the Labour leadership against the defectors is set to continue this weekend, with Corbyn travelling on Saturday to the Nottinghamshire constituency of Anna Soubry, who defected from the Conservatives to the Independent Group this week. She held the Broxtowe seat by 863 votes at the last election.

The Labour leader is expected to say: “I’m disappointed that a small number of Labour MPs have decided to leave our party and join forces with disaffected Tories, who say they have no problem with austerity that has plunged thousands into desperate poverty and insecurity.”

Meanwhile, Labour is edging closer to backing a second EU referendum, with some members of the shadow cabinet led by the shadow Brexit secretary, Sir Keir Starmer, arguing that would significantly ease the concerns of some backbenchers. John McDonnell, the shadow chancellor, said on Friday: “On the people’s vote, we’ve kept it on the table and we’re moving towards that.”

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Play Video 0:49 Ian Austin: I'm quitting because Corbyn has created a culture of extremism - video

Austin, who was a junior minister under Tony Blair and Gordon Brown, has had a series of run-ins with the party leadership since 2015 when Corbyn took over. In 2016, he told Corbyn to “sit down and shut up” and shouted “you’re a disgrace” as the Labour leader criticised the Iraq war in the House of Commons.

Last year, he was placed under investigation by the party after he became embroiled in a row with the party chairman, Ian Lavery, over the handling of the antisemitism issue. The investigation was dropped in November and Austin hit out at the “appalling” handling of the case.

On Friday, he said: “I am appalled at the offence and distress Jeremy Corbyn and the Labour party have caused to Jewish people.”

McDonnell signalled that he agreed with some of Austin’s concerns. “We’ve got to be quicker, and we’ve got to be fiercer,” he said. “I think there’s been a lot of listening but not enough action. That’s the problem.”

Falconer said he had been talking to Formby about taking on the oversight role for a few weeks and was hopeful of being appointed in the next week or two. MPs from across the party said that they would welcome Falconer’s appointment.

Unlike the defectors to TIG, Austin has been supportive of Brexit and was one of three Labour MPs who voted with Theresa May’s government in support of her deal, which was nevertheless rejected by 230 votes. Voters in his constituency backed Brexit by 71% in the 2016 referendum.

But Austin did appear to indicate he could work with TIG in the future. He told BBC WM 95.6 radio: “That’s not what it is about today; I agree with them that the Labour party is broken.”

A party spokesperson said: “We regret that Ian Austin has left the Labour party. He was elected as a Labour MP and so the democratic thing is to resign his seat and let the people of Dudley decide who should represent them.”