Sir Keir Stamer has insisted the "vast majority" of Labour MPs supported him and wanted him to continue doing his job as prime minister.
He was speaking to the Sunday Times at the end of a week in which speculation about a potential leadership challenge has risen.
Sir Keir told the paper "in politics, you get this sort of thing all of the time, there is always talk".
Shadow home secretary Chris Philp said the prime minister should resign over his decision to appoint Lord Mandelson as the UK's ambassador to the US, adding that Labour MPs should "develop a backbone and get rid of him" if he did not step down.
Philp told the BBC's Sunday with Laura Kuenssberg programme that some ministers were "equivocating" over Sir Keir's future and that a couple of MPs were publicly calling on him to go.
He said it was in "the national interest" for MPs to remove the prime minister.
In his interview with the Sunday Times, Sir Keir said "what you never hear from are all the people who are supportive, loyal and just want to get on with the job. And that is the vast majority of people in the Parliamentary Labour Party".
"They're pleased to be in power," Sir Keir said of what he described as most Labour MPs.
"They've waited a long time to be in power. And they just want to get on with their job. They don't make a lot of noise about it. They don't talk to journalists about it. It's really important that is reflected in these debates."
The prime minister has faced calls from across opposition parties to resign and criticism from within his own party after it emerged that UK security officials had flagged concerns about giving vetting clearance to Lord Mandelson.
Sir Keir told MPs this week that the security officials had recommended against vetting approval being granted but that this had not been passed on to him.
He told the Sunday Times that he did not regret his decision last week to sack the most senior civil servant in the Foreign Office, Sir Olly Robbins, who had not informed him about the security vetting concerns.
"When there's a double red flag not to give clearance and [showing] high concern, then I'm sorry. I'm sorry. But I do not accept the argument that that is something which should not be told to the prime minister," Sir Keir said.
Sir Olly Robbins told MPs this week that he had not been told that there was a recommendation to deny Lord Mandelson security clearance, only that officials were "leaning against" it. Sir Olly decided to approve the vetting subject to mitigation being put in place.
Sir Keir dismissed the suggestion he should have done more to inquire about the outcome of Lord Mandelson's vetting process.
"When I'm told there's security clearance, should I go back and quiz officials and say, 'Are you telling me the truth?'
"If I questioned every bit of information put in front of me I would never get anything done. The number of decisions that have to be made each day is huge."
The prime minister told the paper that he wanted to focus on the wars in Ukraine and Iran rather than speculation about possible challenges to his leadership, as the effect of those conflicts for the UK was the most important issue facing the country.
"This is the urgent issue of our time," Sir Keir told the paper. "This is going to reshape our country.
"The conflict with Iran has not just been fought out in Iran. There are increasingly the use of proxies in this country. Of course there's lots of discussion in Parliament about who's up, who's down and all the rest of it. But this is the serious work of being the prime minister."
Philp said the parliamentary Privileges Committee should investigate whether Sir Keir had "misled Parliament" over Lord Mandelson. A new inquiry by the committee would require the approval of MPs in the Commons.
Darren Jones, the chief secretary to the prime minister, told the BBC that the government had "nothing to hide" over the appointment of Lord Mandelson, adding that it was "frustrating that so much time is being spent on this".
He said the upcoming elections in May were "going to be difficult" for Labour and that the "latest round of issues has made this slightly harder".
However, he said that, during his time campaigning, "not one person" had raised the subject of Lord Mandelson.
