A series of senior Labour figures have dismissed calls for a new investigation into what Keir Starmer told MPs about the appointment of Peter Mandelson as political point scoring, before a possible Commons vote on the issue.

The Conservatives have called for the cross-party privileges committee, the remit of which includes examining whether MPs broke rules, to look at whether the prime minister misled parliament when he said normal procedures were followed with Mandelson’s appointment.

The privileges committee previously examined Boris Johnson’s behaviour around lockdown-breaking Downing Street parties during Covid, finding he deliberately misled parliament in saying no rules had been breached.

The foreign affairs committee has already begun an inquiry into Mandelson’s appointment. Downing Street says the evidence it has heard so far, including from senior civil servants, has shown Starmer told the truth.

Before an expected attempt by the Conservatives on Monday to push for a Commons vote on a new inquiry, the former cabinet ministers Alan Johnson and David Blunkett released a joint statement calling the move a “nakedly political stunt with no substance”, calling any comparison to Johnson “absurd”.

“When parliament referred that matter to the privileges committee, a police investigation had directly disproved his categoric statements that he knew nothing about the breach of lockdown rules including parties in Downing Street, and therefore he had a case to answer for knowingly misleading the House of Commons,” they said.

Emily Thornberry, the Labour MP who chairs the foreign affairs committee, said she could not see the need for a second inquiry while the one she is leading was still taking place.

“It may be that at some stage in the future some of the questions haven’t been answered and it is decided that they are of sufficient importance that the privileges committee should be involved,” she told BBC Radio 4’s Today programme.

Thornbery said she did not see the need for it before that stage had been reached “apart from potentially people trying to score points in advance of the local elections”.

Her committee heard last week from Olly Robbins, the former head civil servant at the Foreign Office who was removed by Starmer for failing to tell No 10 that Mandelson had initially failed security vetting, and Cat Little, the lead official at the Cabinet Office.

On Tuesday, it will take evidence from Philip Barton, Robbins’ predecessor at the Foreign Office, and Morgan McSweeney, Starmer’s former chief of staff, who is close to Mandelson and identified as someone likely to have pushed for his appointment.

It will be up to the Commons speaker, Lindsay Hoyle, to decide whether to allow a vote on referring the matter to the privileges committee. If there is a vote, there is speculation that Labour may whip its MPs to oppose the idea.

Emma Reynolds, the environment secretary, who was sent out for broadcast interviews on Monday morning, declined to say how Labour MPs might be told to vote. She called the proposed vote “silly political games” by the Conservatives.

“Ten days ago, we had Kemi Badenoch and other members of the opposition saying that the prime minister deliberately misled parliament,” Reynolds said. “He didn’t, and that was categorically proven last week [in the evidence of Robbins and Little], and they have accepted that. He has not lied to parliament.

“So I do think that the opposition – guess what, 10 days out from local elections and important elections in Scotland and Wales – are playing silly political games when we should be talking about the big issues at stake in the country here.”