As the shadow Cabinet member held a commanding lead in the latest poll in the contest to replace Jeremy Corbyn, Sir Keir insisted the party needed to change to regain the trust of voters.
He denied that the Prime Minister’s force of personality was a key part of the Tory victory, telling ITV’s Peston: “No. We made him look good.
“I think I went to 44 constituencies with campaign teams in the general election – nobody reported back to me the people were saying ‘Boris Johnson’s great, he’s just what we want’.
“What they were saying was we want change, desperately want change, but we don’t trust your party.”
Asked who his campaign was being funded by, Sir Keir said: “There are very strict rules – the Electoral Commission and the parliamentary rules require you to register all your funding and then to publish it, and that’s what we’ve been doing.”
He said the “first tranche went up on the parliamentary website a few weeks ago” and the parliamentary authorities are checking the next tranche. “I think it’s going up as soon as they give it the green light, which will probably be on Monday of next week,” he added.
The comments came as a YouGov poll for Sky News of people eligible to vote in the leadership election put Sir Keir on 53%, well ahead of left-wing candidate Rebecca Long-Bailey on 31% and Lisa Nandy on 16%.
The survey suggests Sir Keir could win in the first round of the ballot.
Mr Khan said Sir Keir is the “best person to unite our party” and put Labour in government.
The Labour mayor, who was MP for Tooting before entering City Hall, is one of the party’s most powerful elected politicians.
Announcing his endorsement on Twitter, he said: “I will be voting for @Keir_Starmer to be the next Labour leader. I’ve known Keir for decades.
“He’s the best person to unite our party, take the fight to the Tories and put Labour in government. #AnotherFutureIsPossible.”
Sir Keir, the shadow Brexit secretary, said he was “honoured” to have received Mr Khan’s backing.
“Sadiq is demonstrating that Labour in power can change lives. I look forward to campaigning alongside him to keep London Labour.”
Members and party supporters have begun voting in the contest this week, with batches of ballot papers being sent out by email and post.
They will also select a new deputy leader from Angela Rayner, Dawn Butler, Ian Murray, Dr Rosena Allin-Khan and Richard Burgon.
The winners are set to be named on April 4 and will take up the roles immediately.
PA Media contributed to this story