Keir Starmer has said he will lead Labour into the next general election as his Downing Street allies denied claims of any wrongdoing over the appointment and vetting of Peter Mandelson as US ambassador.

Darren Jones, the chief secretary to the prime minister, told Sky News’s Trevor Phillips on Sunday show that “no wrongdoing by the prime minister has been proven in relation to Lord Mandelson’s appointment”, adding “the whole situation is regrettable”.

Jones said: “Of course these particular questions of whether the prime minister had lied or misled the house, or had done any wrongdoing here that’s all been shown not to be the case.”

It has been a bruising week for the prime minister after the Guardian revealed that he appointed Mandelson despite vetting officials recommending that he be denied security clearance. His handling of the row was called into question, including his swift decision to sack the Foreign Office chief Olly Robbins.

On Saturday night, Starmer attempted to brush off mounting unrest inside his party and shut down questions about the future of his leadership in an interview with the Sunday Times. Asked directly if his prime ministership was over, Starmer told the newspaper: “No.”

Pushed on whether he could continue, he replied: “Of course. We didn’t wait 14 years to get elected, we didn’t change the Labour party, we didn’t do all that it entailed to win the election and the mandate for change, not to deliver on it.”

Asked if he would lead Labour into the next election, Starmer said: “Yes.”

The Conservatives have sought to corner the prime minister over the row, highlighting inconsistencies between Starmer’s previous claim that “no pressure whatsoever” was applied to Robbins, and the former senior civil servant’s describing “an atmosphere of pressure” and “constant chasing” from Downing Street while Mandelson’s vetting was taking place during an interview with the foreign affairs committee.

Starmer rejected the claims, saying “no pressure existed whatsoever”. He added: “There’s pressure – ‘Can we get this done quickly?’ – which is not an unusual pressure. That is the everyday pressure of government.”

Starmer also rejected suggestions that he as the prime minister should have been more inquisitive and curious about Mandelson’s clearance. “When I’m told there’s security clearance, should I go back and quiz officials and say, ‘Are you telling me the truth?’” Starmer added.

The Conservatives are also pushing for Starmer to face the Commons privileges committee over allegations that he misled parliament.

The revelation that Starmer had appointed Mandelson despite officials recommending he be denied security clearance came about because of a process started by Kemi Badenoch, the Conservative leader.

Chris Philp, the shadow home secretary, told the BBC’s Sunday with Laura Kuenssberg show: “Now we need the privileges committee to investigate whether Keir Starmer misled parliament, which he appears to have done.

“Frankly Keir Starmer should resign because he appointment Mandelson in the first place, which is a terrible, terrible misjudgment – a man with links to Russia and China and Jeffrey Epstein.”

However, the former Tory attorney general Dominic Grieve accused his party of “political games” by seeking a vote on launching a privileges committee investigation. Grieve said: “The appointment of Peter Mandelson is currently being properly scrutinised as part of the humble address process and the work and hearings being conducted by the foreign affairs committee.

“Seeing that this appointment is being explored during these processes and all the evidence is not yet considered, it is impossible to see how starting a reference to the privileges committee is either useful or indeed ‘due process’.

“Ensuring that ministers do not mislead parliament is of great importance but it should not just be turned into political games.”

Harriet Harman, who chaired the privileges committee investigation into Boris Johnson, said it would be a “complete duplication” of the work being done already by the foreign affairs committee.

Questions over Mandelson’s vetting will continue into the coming week,with Starmer’s former chief of staff Morgan McSweeney due to appear before the foreign affairs committee.