Labour leadership hopeful Andy Burnham has said he is "confident" he will have enough support to get on the contest's ballot paper on Wednesday.
The former health secretary needs the backing of 33 Labour MPs - he has 21.
David Miliband has denied being a "bottler" by not challenging Gordon Brown earlier, saying he did not want to "make a bad situation worse".
And rival Ed Balls has said Labour was wrong not to introduce migration controls on new EU states in 2004.
Labour leadership contenders have until Wednesday to get 33 colleagues to back them. MPs, MEPs, party members and members of affiliated organisations, such as trade unions, will then get a vote in the leadership election in September.
So far only Mr Miliband, his brother Ed and fellow shadow cabinet minister Ed Balls have sufficient nominations to get through.
But Mr Burnham, who will need to get another 12 MPs' backing in three days, told Sky News: "I'm confident I can reach the target."
David Miliband, who is backed by 62 MPs, told BBC One's Andrew Marr programme: "If another candidate gets to 32 nominations and needs one extra to get onto the ballot paper, I will give them mine."
Diane Abbott, the only woman standing, has complained that the contest is being drawn from the "narrowest gene pool in history" and criticised the "ridiculously high threshold" of 33 MPs.
She has the backing of seven MPs, while left-wing backbencher John McDonnell has 10.
The leading candidates are all white men with degrees from Oxford and Cambridge, in their 40s, who have worked as special advisers. But Mr Burnham told Sky: "People keep saying everyone's background's the same, but mine is different."
He added: "There wasn't much privilege about going to a Merseyside comprehensive in the 1980s, I can assure you of that."
He said he would be setting out his leadership agenda on Monday, drawn on his own experiences, which would be about "ending uneven life chances in this country, giving more hope to young people".
David Miliband rebuffed suggestions on the Andrew Marr programme that he was the "Blairite" candidate.
The former foreign secretary, who was head of the Downing Street policy unit under Tony Blair before becoming an MP in 2001, said: "I am the Milibandite candidate - if you don't mind, or the Miliband D candidate.
"I think that it's very important that we have a political process where people define themselves by who they are."
He said there was a "range of candidates" with "distinctive talents" and said he had not flinched, in 2008 and 2009, by not challenging Mr Brown for the leadership before the election.
"We all have to take our share of the responsibility but I'm not going to make a bad situation worse and that's what I was being asked to do last summer and the summer before.
"I'm a battler for the things I believe in, not a bottler."
Meanwhile in an interview with the Observer, former schools secretary Ed Balls said high levels of immigration under Labour had had an impact on the pay and conditions of "too many people".
He said there had been "real economic gains" from eastern European immigration but added: "As Labour seeks to rebuild trust with the British people, it is important we are honest about what we got wrong.
"In retrospect, Britain should not have rejected transitional controls on migration from the first wave of new EU member states in 2004, which we were legally entitled to impose."
On immigration he told BBC One's Politics Show immigration should not be seen as a race issue and said he told Gordon Brown over the past 18 months that Labour had been "making a mistake by brushing it under the carpet".
He said Mr Brown's answer to Gillian Duffy - the Rochdale pensioner he was recorded calling a "bigoted woman" - showed "he had not been having the conversation because what she said was the kind of things being said by Labour supporters".
Mr Balls said he supported Turkey joining the EU but said "the kind of transitional arrangements on labour which we should have done to Poland and other A8 countries in 2004 we absolutely must do for a sustained period" as otherwise people's support for the EU could be undermined.
Four of the six hopefuls for the Labour leadership - Mr Balls, the Miliband brothers and Mr Burnham - have agreed immigration was an issue that Labour did not adequately address during the election.