Sadiq Khan has warned that Labour cannot rely on its "core vote" to win elections ahead of a meeting with party leader Jeremy Corbyn.
The new mayor of London warned that appealing to "natural Labour voters" alone would not be enough to secure success at a general election.
Mr Khan secured an overwhelming victory over Tory Zac Goldsmith following a bitter campaign which saw his rival attempt to link him to Islamic extremists.
The Labour mayor said the Conservative campaign was "straight out of the Donald Trump playbook".
Asked on BBC One's The Andrew Marr Show 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 because what it showed was London chose hope over fear and unity over division.
"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."
Mr Corbyn was notably absent from the ceremony on Saturday which saw Mr Khan signed in as mayor - an event attended by previous Labour leader Ed Miliband.
Mr Khan used an Observer article setting out the lessons from his campaign to issue a clear message to the Labour leader that the party needed to broaden its appeal and reach out to Tory voters.
He said that "it should never be about 'picking sides'" - strikingly similar language to the "Elections are about taking sides" slogan used on the party's posters in the local election campaign.
Asked if he was directly attacking Mr Corbyn's campaign strategy, Mr Khan said: "My point is this. I want a big tent, you know, whether you're a Conservative trying to be the mayor of London, or a Labour Party trying to form the next government, we've got to speak to everyone.
"There's no point in us just speaking to Labour voters, our core vote."
Mr Corbyn, who travelled to Bristol to celebrate with that city's new mayor Marvin Rees rather than attend Mr Khan's ceremonial event, has insisted the pair are "getting on fine".
Mr Khan said: "I think we're seeing each other tomorrow."
In his Observer article, Mr Khan said David Cameron and Mr Goldsmith had sought to "divide London's communities in an attempt to win votes".
"They used fear and innuendo to try and turn different ethnic and religious groups against each other - something straight out of the Donald Trump playbook. Londoners deserved better and I hope it's something the Conservative Party will never try to repeat."
Although Labour's council election results were better than many had predicted, it was relegated to third place in Scotland and lost ground in Wales, where it remains the largest party.
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."
Shadow cabinet minister Emily Thornberry defended Mr Corbyn's decision not to attend Mr Khan's signing-in ceremony because he went to Bristol and "you can't be in two places at once".
She told Sky News' Murnaghan programme that Mr Corbyn would agree with Mr Khan's call for Labour to be a "big tent" party.
The shadow defence secretary said: "We've got to make sure that we stop talking amongst ourselves about ourselves and gazing at our navels and we've got to be able to get out there and fight some elections and speak to people about issues that matter."
She also defended Mr Corbyn over his reported plans to take a holiday at the end of the month as the European Union referendum campaign enters its final few weeks.
"Well I think it's been pretty full-on for Jeremy, hasn't it, since he was elected as leader," she said. "I do think that when Parliament is not sitting he should be allowed to have a few days off.
"We will be campaigning full out to stay in Europe, Labour is a united party on this issue and we will be speaking with one voice. It is a team effort and for Jeremy to have a few days off I don't think is anything that anyone should begrudge him."
A spokesman for Mr Corbyn said: "We don't discuss Jeremy's movements but he will undertake a significant number of public engagements with the Labour In campaign in the next few weeks."
Shadow housing minister John Healey told the Murnaghan programme Mr Corbyn had to have a "mission to win" elections.
He said: "Every new party leader has to prove himself. It's only eight months since he was elected.
"But for my money, that mandate to lead must also be a mission to win and it's that single-minded job now, our sole purpose, must be to win back a hearing and then the support of the millions of people in Scotland and across swathes of middle and southern England and Wales that deserted us and weren't convinced by us in 2015."
Former home secretary Alan Johnson told BBC Radio 4's The World This Weekend the "Super Thursday" contest had been "no triumph, no disaster, it was kind of a stand still election".
George Osborne defended Mr Goldsmith's controversial campaign but said Mr Khan had answered the questions posed to him.
The Chancellor told ITV's Peston On Sunday: "Politics is a robust thing in a democracy and (in) elections there's a lot of rough and tumble."
He went on: "You are asked who are you, who do you associate with, what are your ideas.
"But if you can answer those questions, and clearly Sadiq Khan did answer those questions to the satisfaction of London voters, you get yourself elected.
"But that is the process we go through to make sure people are up to the jobs."
Labour MP Clive Lewis, a close ally of Mr Corbyn, denied there was a rift between the leader and Mr Khan, and said the pair were "good friends".
He told BBC One's Sunday Politics: "Sadiq didn't win in spite of Jeremy Corbyn or because of Jeremy Corbyn, he won because he worked with Jeremy Corbyn, all the voters and the people in London that are supporting the Labour Party under Jeremy Corbyn, and by reaching out.
"That's what I take from this and I'm sure that Jeremy and Sadiq will be meeting in the very near future. I don't think it's a big issue, that it's some kind of big division between Jeremy and Sadiq, I really don't."
Mr Lewis said Mr Corbyn is beginning to realise Labour needs to win over more Tory voters.
"We've got to start reaching out to people who in the past have voted Conservative," he said.
"That is a message that Jeremy Corbyn needs to listen to and I think it's a message that he will listen to."
But Labour former minister Caroline Flint insisted the party needs to make "a hell of a lot more progress".
She told the programme: "John McDonnell said in the run-up to these elections 'we're looking to hang on'.
"Looking to hang on isn't enough, this is the worst result for an opposition party after a general election in 30 years.
"The year after Michael Foot was elected I think we gained a thousand council seats, Tony Blair 1,800, Ed Miliband over 800. We actually had a loss of 28."
Shadow cabinet minister Ian Murray told BBC Radio 5 Live's Pienaar's Politics: "We have to build a big tent. Jeremy Corbyn has a significant mandate within the party membership of the Labour Party, he now needs to extend that mandate both to the PLP but (also) to the wider public."
He suggested that Mr Corbyn and shadow chancellor John McDonnell would have to "move their positions to a more pragmatic approach".
"The Labour Party can't get into power in 2020 without attracting votes back from the Scottish National Party in Scotland, without attracting Conservative voters - who voted Conservative in 2010 - back," he said.
Asked about Mr Corbyn's holiday plans, Mr Murray said: "'Given Jeremy Corbyn's significant mandate within the Labour membership and given it's going to be Labour members who are knocking on doors to make those arguments, we should be making sure we are pulling out all the stops to win that European Union referendum."
He added: "We should be fighting with every single sinew to make sure the UK stays in the European Union, because that's the right thing to do."
Shadow chancellor Mr McDonnell, seen as a potential successor to Mr Corbyn, insisted he had no ambitions for the leadership.
Mr McDonnell told BBC Radio 4's Westminster Hour: " Jeremy is my closest friend in politics over 30 years. What do I have to do? Have a civil partnership with him or something? It's ridiculous."
He added: "I have waited most of political life for a socialist to be leader of the Labour Party and here he is and I am doing everything I can to support him."
Mr McDonnell vowed that he and Mr Corbyn would unite the party: "When Jeremy was elected he invited everybody who was in the existing shadow cabinet to come and play a role including people like Caroline Flint and others and he created that big tent and some of them have just excluded themselves from offering their services to the party and that's up to them.
"That's fair enough. Of course people can express their dissent and we can have that policy debate and then we'll arrive at a democratic decision and then we expect people to unite in the interests of the party and the country overall what to what we don't expect is people going to the media two days before polling to say there is going to be a coup against the leadership. That is not the loyalty we'd expect from any element of the party.
"However we will move on from that. We'll unite the whole party. Every one of those dissenters will be invited in to see Jeremy, they will offered roles in the campaign in the future and in our administration.
"Everyone has got the opportunity to serve and we hope then we can unite the party and move on and campaign now into the referendum campaign."
He said Mr Corbyn deserved a break: "If he wants a couple of days off I don't blame him because at the moment he seems to be working seven days a week."