Labour leadership hopefuls Liz Kendall and Yvette Cooper have said their supporters should back anyone other than Jeremy Corbyn in the contest.
Ms Kendall told the BBC Labour risked sending a "resignation letter to the British people as a serious party of government" by electing Mr Corbyn.
Separately, Ms Cooper warned there was a "serious risk the party will split" if the left-winger becomes its leader.
It comes as Labour begins sending out the first ballot papers to voters.
The result of the contest will be announced at a special conference on 12 September.
More than 600,000 people have signed up to vote in the four-way contest but Labour has said applications are still being verified.
Meanwhile voting in the election for the new Scottish Labour leader ended at midday.
Mr Corbyn is due to unveil a 10-point policy plan while in Glasgow later.
The popularity of the left-wing Islington North MP, who is promising "a new kind of politics", has sparked a row about the future direction of the Labour party.
Another leadership contender, Andy Burnham, told the BBC Mr Corbyn's policies "lack credibility".
"It's not possible to promise free university education, re-nationalising the utilities, without that coming at a great cost and if you can't explain how that is going to be paid for then I don't think we'll win back the trust of voters on the economy," he said.
BBC political correspondent Ross Hawkins said there had been "frustration" in rival camps who accused Mr Burnham of being reluctant to take on Mr Corbyn. This appeared to be his most direct attack yet, he added.
But in an interview with Jeremy Vine on BBC Radio 2, Mr Burnham declined to follow Ms Kendall and Ms Cooper and advise his supporters not to back Mr Corbyn with their second and third preferences.
He added: "People will say if they hear things like that, 'hang on, what do you believe?'"
In an interview with The Independent, Ms Kendall called for voters to mark Ms Cooper or Mr Burnham as second and third preferences, and avoid giving votes to Mr Corbyn.
"I have set out very clearly where I differ with all the candidates but our differences with Jeremy's kind of politics are far greater," said Ms Kendall.
Speaking on BBC Radio 4's Today programme she said she "can't pretend to be agnostic" about a victory for Mr Corbyn, saying of the voting process: "It is an alternative vote system and I want to urge party members to use all of their different preferences.
"I will be using my second and third preferences and I would urge others to do the same because I don't want to see our party go back to the politics of the '80s, just being a party of protest."
The Leicester West MP also said she did not see the party splitting, as it did in the 1980s when Labour members formed the Social Democratic Party.
However, Ms Cooper told BBC 2's Newsnight: "I think there is a serious risk that the party will split, will polarise and I cannot bear to see that happen because there is too much at stake."
Asked in an interview with grassroots Labour website Labourlist whether voters should use their votes to try to prevent Mr Corbyn winning, she said: "I think people should use all of their preferences.
"And I think the focus has to be how do we make sure we can win that election, and that's the most important thing - and I don't think Jeremy can do that."
Mr Corbyn has warned against "personal abuse" in the campaign, saying he wants to focus on policy.
His policy programme includes a commitment to "growth not austerity", nationalising the railways and energy sector, and a plan for nuclear disarmament.
In an essay for the Fabian Society he also suggested Labour's new increased following should be more involved in the party and proposed a review of membership fees to make the party more "inclusive".
Former Prime Minister Gordon Brown is expected to join the debate over the leadership contest with a speech on Sunday, called "power for a purpose - the future of the Labour Party".
Lance Price, former director of communications for Labour, told the BBC the contest had been an "unedifying mess" and had "done nothing to reengage the labour party with those millions of people who deserted it".
The Guardian newspaper has endorsed Ms Cooper for the leadership while the Daily Mirror has given its backing to Mr Burnham, although the paper urged him to "find a role" in his team for Mr Corbyn, who it says has "lit up the election campaign".