Andy Burnham, the shadow health secretary, has made a surprise move to assert his independence from the trades unions by saying he will not take any union cash to fund his Labour leadership campaign. Burnham made his announcement as one of his chief rivals, Liz Kendall, tried to seize the mantle as the change candidate and new Labour MPs expressed their concern that the nomination process may narrow the field to two before the party has had a chance to hear the start of a debate.

Burnham has been labelled in some quarters as the unions’ favoured candidate, but he said: “I am not going to take any money from the trades unions in this leadership campaign. No money has been offered, but if it was, I would encourage it to be given to the Labour party to assist the rebuilding after the election. But I am actively seeking the support of individual trade union members and am pleased they have a bigger say in this contest.

“I am aware that, whatever the result of this contest, the party must come out of this well. I am going to be my own man. I am independent and will make my own judgments. I make no apology for our historic links with the trades unions.

“What I have decided is a risk, but it is a sign of my confidence and openness that this contest is conducted in the right way. If I take no money from the unions, I am not dependent on them, and that will put me in a stronger position to defend unions and to defend the union link in the future, which I intend to do.” Candidates are facing a spending cap that could be as high as £300,000.

Two new Labour MPs have urged the parliamentary party to ensure that the nomination process does not lead to the contest being narrowed to two candidates, likely to be Burnham and the shadow home secretary, Yvette Cooper. The letter from Stephen Kinnock, the MP for Aberavon and son of the former party leader Neil Kinnock, and Jo Cox, the MP for Batley and Spen, expressed fears that “the field may end up being dramatically narrowed before members and registered supporters have had a chance to cast a single vote”.

The pair write: “As newly elected MPs, we are being asked to nominate candidates on the basis of very little information about how they plan to rethink, rebuild and relaunch our party. The primary purpose of the nomination phase should be to ensure that Labour party members and registered supporters across the length and breadth of the UK are ultimately presented with a strong and diverse shortlist.”

They urge any candidate that has reached the threshold of 35 nominations to encourage some of their surplus nominators to make a tactical transfer to contenders who are struggling to reach the threshold. “We want our members and supporters to be faced with a range of voices and ideas, not a shortlist of one or two. This diversity will improve the quality of our internal debate, and will ultimately improve the decisions we take.”

It is understood that Burnham is willing to consider transferring a surplus nomination from one of his supporters if a candidate is just short of the 35 required to get into the field.

The two critical unallocated nominations in the parliamentary party are those of Chuka Umunna, the shadow business secretary, and Alan Johnson, the former cabinet minister. Johnson has described himself as a floating voter. Umunna, who pulled out of the running at the beginning of the week, has been courted by supporters of Cooper and is said to be undecided, but was impressed by a hard-hitting speech delivered to the press gallery at Westminster on Thursday.

Kendall warned the party not to cling to the “fantasy” that Britain had swung to the left as she backed successful free schools and pledged to fight defence cuts. She said Ed Miliband’s flagship energy price freeze had not been believable, that she would not prioritise cutting tuition fees, and that the party should back any successful school “regardless of whether it’s a local authority, academy or free school”.

She argued that the contest “cannot be about who the [union] general secretaries say impresses them the most”. “When it comes to public services, I am firmly on the side of the public,” she said. “Services should revolve around those who use them and be fit for the future, not stuck in the past. There is no point saying you believe in economic responsibility and being careful with taxpayers’ money if public services are a reform-free zone.

“When Labour loses, we do one of three things. We decide we didn’t win because we weren’t leftwing enough – fantasy. We decide we can avoid the really tough decisions because they are too uncomfortable – a fudge. Or we decide that winning is too important.

“Everyone in this contest will claim to be the candidate of change. It’s progress of a sort that they feel they need to. But this election isn’t about claiming to change. It’s about who will make sure Britain meets the challenges of the future.”