Jeremy Corbyn has conceded he will have to win over people who have previously been "tempted to vote Tory" to have a chance of forming a Labour government.
The Labour leader stressed the importance of engaging young people and energising the party's supporters to win the next general election.
But he admitted during the party's latest leadership debate that Labour will have to appeal to people who have previously considered voting for the Conservatives.
His challenger for the leadership, Owen Smith, went further and said Labour needs to win over two million people who voted Tory at the 2015 general election.
Speaking at the event in Birmingham, Mr Corbyn said: "We win over people to support us by the policies we put forward.
"Yes, we do have to win over some people that have been tempted to vote Tory.
"We win them over on the basis that we can create a society where we do provide decent housing for all, where we do provide real security for all in work.
"We also say to those people who are relatively well off, 'are you happy walking past the homeless and the starving on the streets of our country when there is no need for that, when we could do things very differently?'.
"But we also, I think, win an election by inspiring our own supporters, inspiring those that have supported other parties, but above all reach out to young people in our society, only 47% of whom voted in the last election to come on board with us and try and create that decent society."
Mr Smith replied: "I don't just think we have got to win over some people who have been tempted to vote Tory, as Jeremy put it. I think we have got to win over two million people who voted Tory just a few months ago.
"We got nine million, they got 11 million. I want a Labour government in order to put into practice my principles and in order to do that we have got to get the best part of two million people who voted Tory to vote for us."
There are due to be nine official leadership debates between the pair as they battle it out for the party's top job.
The event in Birmingham was the fourth.
It came a day after Mr Smith was accused of being unfit for high office when he suggested Islamic State would have to be brought into peace talks to end the Syrian civil war.
Mr Smith was jeered by some members of the audience as he referred to "170 socialist MPs" who lack confidence in Mr Corbyn's leadership.
Debate moderator Carl Dinnen, of ITV News, sought to calm tensions while Mr Smith said: "This is what is going wrong with the Labour Party. We are beginning to look at one another as though we are each other's enemies, we are beginning to treat one another in a deeply uncomradely fashion.
"The suggestion that Jeremy is the only socialist in our party is an insult to many of us who are socialists. We're not red Tories or anything else that we're branded on Twitter, we're people who want to create a more equal Britain."
Mr Corbyn asked the audience for a debate in a "comradely manner", adding he was "very disappointed" when Mr Smith and others resigned from the shadow cabinet and called for a leadership change.
He said he will seek to "reach out" if re-elected as leader to develop a "coherent, cohesive" Opposition.
Mr Smith claimed Mr Corbyn only met him once during his nine months as shadow work and pensions secretary, with the incumbent leader countering that they met every week in the shadow cabinet.
On the prospect of Republican nominee Donald Trump becoming US president, Mr Smith said: "I'd be prepared to work with whoever the president was of America, that would be in the interests of the British people and we want a serious Labour government that would always act in the interests of the British people."
Mr Corbyn criticised Mr Trump's "simplistic statements" as being "quite often beyond belief", adding he hopes he is not elected president.
He also said he "would have to negotiate with the USA" should Mr Trump be elected.
Mr Smith reiterated his strong support for keeping the UK in the EU, telling the debate: "I would fight for us to stay in the European Union. Every day I was the leader, I would fight for it."
Mr Corbyn in his remarks noted: "We've got to recognise a referendum took place and while the result wasn't one either Owen and I wanted, it was a result we obtained or achieved or was there.
"I want to say we have to now negotiate to protect environmental conditions, protect consumer rights, protect workers' rights that have been gained through Europe, we have to maintain a market access to Europe, maintain membership of the European Investment Bank and a trading relationship with Europe."
Mr Corbyn and Mr Smith were also asked if they would abolish grammar schools amid rumours the Government is considering allowing new selective schools to open.
Mr Corbyn said: "I would want to create a comprehensive education system for everybody.
"I would want those grammar schools to become comprehensives, although I recognise there's a huge battle to achieve that."
Mr Smith said he "absolutely wouldn't extend them".