Left-winger Jeremy Corbyn would lead Labour to "oblivion", an ally of leadership rival Liz Kendall warned as the contest became increasingly bitter.
Shadow education secretary Tristram Hunt said Labour had to face a "summer of hard truths" and must not wrap itself in the "comfort blanket" of the left.
In a further sign that the other contenders are worried about support for Mr Corbyn, Yvette Cooper warned against becoming a "narrow party of the left".
Mr Corbyn has secured the backing of leaders of the country's biggest trade union Unite and trails only frontrunner Andy Burnham in nominations from constituency Labour Parties.
Mr Hunt, a prominent supporter of fellow moderniser Ms Kendall, told BBC1's Andrew Marr Show: "Jeremy is a long-standing figure within the party. But what I would say to members and colleagues is that if we seek a comfort blanket we are going to be condemned to oblivion, because Jeremy is not asking the hard questions about economic competence.
"One of the areas where we were hammered in the general election was that people did not trust us with the public finances, people did not trust us in an era of austerity.
"I don't think Jeremy Corbyn has the answers to that."
Shadow home secretary Ms Cooper told Pienaar's Politics on BBC Radio 5 Live: "I think in the end, most people know in the party that we're not going to win by turning the clock back; we've got to be looking forward to the future. I don't think the right answer for the country is to move to a narrow party to the left."
Acting Labour leader Harriet Harman said she would stay out of the battle to replace Ed Miliband, but urged the party to elect someone who could "be a leader for the country as a whole, not just for the comfort of the Labour Party".
She told BBC1's Sunday Politics: "The point is not to have somebody that we particularly like and feel comfortable with; the point is to have somebody who can command the confidence of the country, and that's what they should have in their mind.
"There's no point doing choice in a disappointed rage. We've got to actually be doing a sensible choice for the future."
Reflecting on the party's defeat, she said there was a sense that people felt it was a risk to vote Labour: "One of the saddest things is somehow even people who supported us, some of them felt, well a few, perhaps actually it is a bit of a relief that Labour didn't get elected."
Mr Corbyn insisted that Labour was a "very broad church" able to accommodate his views and those of Blairites such as Mr Hunt.
He told Sky News' Murnaghan programme: "I understand there are huge differences and debates.
"But what unites the party is a hatred of inequality, a hatred of injustice, a determination that we should provide housing for everybody, a determination that our young people should get work, our young people shouldn't be saddled with unsustainable debts when they leave university."
Mr Corbyn said the party had fought the last election under Ed Miliband on an "austerity-light" platform.
He said: "I think what we need to say is, let's not choose an arbitrary date to go into budget surplus, let's instead look at the issues of the question of poverty, the question of the waste that poverty brings about, and the need for economic expansion rather than an arbitrary date for budgetary surplus.
"Even the USA has gone for investment rather than austerity and they have much lower rates of unemployment than we do. Other European countries are beginning to do the same.
"So surely the lessons from Greece, Spain, Italy and Portugal is that if you go for austerity-heavy what you end up with is a declining economy, higher levels of unemployment, lower levels of tax income and then the situation goes on a worse and worse downward spiral."
Mr Burnham indicated that he would not give any "special favours" to The Sun newspaper, telling the BBC he had not forgiven the paper for its coverage of the 1989 Hillsborough disaster.
Asked specifically about The Sun, Mr Burnham told the BBC's Sunday Politics North West : "I give interviews generally and people can report my words. But I don't do special favours for newspapers that attack me and attack my party."
But Mr Hunt said: "Lots of Sun readers vote Labour, we need to reach out to those voters."