In highly pointed remarks Mr Khan, who on Friday became the country’s most powerful directly elected leader, said Labour under Mr Corbyn was simply not doing enough to address the concerns of ordinary voters.
And he warned that unless Mr Corbyn changed tack and reached out to the whole electorate – not just natural Labour supporters – then party’s central mission to improve the lives of ordinary working people would be put in jeopardy.
“In Labour our mission is to improve the lives of people,” he told The Andrew Marr Show.
“We only do that by winning elections. We only do that by speaking to people who have not voted Labour. There is no point just speaking to Labour voters – our core vote – we need to speak to everyone.”
Pointedly Mr Khan has yet to meet Mr Corbyn since he was elected Mayor late on Friday night. Some around the Labour leader are concerned that Mr Khan intends to use his new office, and the national platform it gives him, to set up a rival power base to Mr Corbyn.
Asked how whether he owed some of his election victory to Mr Corbyn, Mr Khan replied: "Success has many parents and I think what's important is the victory on Thursday was a victory for London. My point is very simple, we've got to stop talking about ourselves and start talking to citizens about the issues that matter to them."
His comments in the last 48 hours will have done little to allay those fears. In an article for the Observer Mr Khan, who ran under the slogan "a Mayor for all Londoners" added that the party needed a broader reach.
“Squabbles over internal structures might be important for some in the party, but it is clear they mean little or nothing to the huge majority of voters,” he wrote.
“As tempting as it might be, we must always resist focusing in on ourselves and ignoring what people really want.
"It should never be about 'picking sides', a 'them or us' attitude, or a having a political strategy to target just enough of the population to get over the line. Our aim should be to unite people from all backgrounds as a broad and welcoming tent – not to divide and rule.”
Mr Khan was backed by the former Labour Cabinet Minister Lord Blunkett who said party members were “kidding” themselves if they felt the performance in last week’s elections were adequate.
“The whole Labour project under Jeremy Corbyn and his allies is flawed,” he wrote in The Sun on Sunday. “They seem to think we won’t have to win back the support of those who voted Conservative last time to gain power in 2020.”
But the deputy Labour leader Tom Watson dismissed the prospect of Mr Corbyn facing a challenge and pleaded for "patience" after a "mixed bag" of election results.
Writing in the Sunday Mirror he said a leadership challenge was "about as likely as a snowstorm in the Sahara".
But he acknowledged: "The truth is Labour still has a mountain to climb if we are to return to government in 2020."
He said: "If there is one quality Labour Party members will need as we seek to return to Downing Street it is patience.
"Jeremy Corbyn was elected leader of our party eight months ago with an overwhelming mandate to take the party in a new direction.
"But that won't happen overnight. Our share of the vote was higher than it was a year ago, when we suffered a painful election defeat.
"Of course it isn't enough. We need to do far more. We need to do better.
"I have been a member of the Labour Party for well over 30 years and I know that members are fair-minded people.
"That's why a leadership challenge is about as likely as a snowstorm in the Sahara."
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