Ken Livingstone, the former Labour mayor of London, has said the grassroots group Momentum should be free to push for changes to Labour party structures that would secure Jeremy Corbyn’s legacy as a leftwing Labour leader.

The party’s deputy leader, Tom Watson, had accused the radical left group of “entryism” after its chair, Jon Lansman, was secretly recorded at a local meeting describing plans to elect a raft of leftwing candidates to key positions and his hope to seal an affiliation from trade union Unite.

A day of public accusation and backroom briefing by Watson and Corbyn allies ended in a fiery private meeting of MPs and peers on Monday night in parliament, where MPs clashed with Corbyn over Momentum’s influence on the party.

The tension hinges on a clause that Corbyn allies hope to secure at the next Labour party conference, to reduce the threshold of MP nominations needed for the next Labour leadership elections from 15% of MPs to 5%, which would make it easier for a leftwing successor to Corbyn to make it on to the ballot paper sent to members.

The Islington MP was elected after a number of MPs “lent” him their nominations in order to have a broad church of candidates, but moderates would be unlikely to do so again for any other candidate allied with Corbyn.

Livingstone, who is suspended from the Labour party after an antisemitism row where he said Hitler had supported Zionism, said MPs should have no issue with the changes.

“I just find it bizarre because what Momentum wants to do is go back to the rules the Labour party had for 90 years when there’s a leadership election, someone from the left can stand,” he said. “We never had this nonsense that you have to get 15% of Labour MPs to nominate you.”

Livingstone said he was happy with the influence of Momentum on the Labour party. “Don’t try and block Momentum wanting to go back to the rules we had, that allow a leadership election to be open,” he said.

“It would be ridiculous to keep a rule that means if Jeremy Corbyn was run over by a bus tomorrow we wouldn’t be able to have a leftwing candidate stand. They wouldn’t get the 15% of MPs. In America, anyone can run for president, they don’t have to get the 15% of members of Congress.”

Livingstone said the unpopularity of Labour in recent polls, currently about 19 points behind the Conservatives, was “not Jeremy’s fault, it’s people who undermined him for the last 18 months” and said the parliamentary party had “skewed to the right” under Tony Blair.

“Jeremy Corbyn doesn’t give in to this sort of pressure. He’s been consistent and however much the press ramp it up, or he’s stabbed in the back by Labour MPs, he’s not going to change what he believes or the direction he’s taking,” Livingstone said. “We now have to focus on getting a good economic strategy out there that can win the next general election.”

Corbyn released a short recorded clip on Monday night calling for unity and insisting that Labour could still be a force in British politics.

“Sometimes spirits in the Labour party can run high. Today has been one of those days,” he said. “My plea to all Labour party members, whether grassroots or in senior positions, is: think of our people, first; think of our movement, first; think of the party, first,” he said.