In a dramatic bid to outflank the Prime Minister, Mr Burnham declared he wants to change the Opposition’s policy to support a national poll on the key issue.
He also warned Mr Cameron he would be “held to account” if he failed to negotiate a good settlement with Brussels that addressed concerns over immigration.
The intervention follows calls from Tory backbenchers for the referendum to be held earlier than 2017, as the premier has promised.
It comes as Labour figures brace for tension with the unions after Jim Murphy announced he was quitting as the party’s leader in Scotland and delivered a stinging parting shot at Unite boss Len McCluskey.
Mr Murphy, who resigned despite winning a confidence vote, warned that Mr McCluskey was the “kiss of death” for Labour and condemned his “destructive behaviour”.
Mr McCluskey is set to respond in interviews scheduled later.
In an interview with the Observer, Mr Burnham said: “The country has voted now for a European referendum and under my leadership the Labour party will not be a grudging presence on that stage. We will now embrace it. It should be brought forward to 2016.
“It should be in the Queen’s speech that it should be in 2016, and the message I would send to Cameron is that I would offer support to deliver it in 2016. It is not going to be in anybody’s interest for this to rumble on through this parliament.
“We have to get to it. We have to do it, embrace the argument. That is the most fundamental problem facing British business right now.”
Mr Burnham, who made clear he would campaign for an “in” vote, went on: “If Cameron doesn’t deliver legislative change in terms of abuse of the rules of free movement by agencies and the effect on people with jobs here, it won’t be good enough. It really won’t be good enough.
“I am passionately pro-European. I cannot see how it could possibly be in the interest of this country to come out of the European Union. This is the challenge that prime minister has set himself and he has to deliver.”
Earlier, the Leigh MP joined other declared candidates, Yvette Cooper, Mary Creagh and Liz Kendall at a hustings in central London with Tristram Hunt, who is expected to put his name forward next week.
Mr Burnham said despite a disastrous election night that saw Labour almost wiped out in Scotland, things could still get worse.
“I do not assume that this is Ground Zero for the Labour Party, that things can only get better,” he said.
All the contenders set out positions conceding Labour should have handled the public finances differently before the credit crunch.
In one of the strongest admissions, Mr Hunt said the party had been spending to “fix the roof while the sun was shining”.
But he added: “I think that we should admit that we spent too much in the last Labour government ...
“What we did not do was leave enough headroom to deal with the financial crash.”
He said the party had allowed the economy to become too reliant on financial services, and called for spending priority to be given to early years education and training in order to boost productivity.
Meanwhile, shadow business secretary Chuka Umunna has flatly denied the shock decision to drop his Labour leadership bid was due to a secret scandal in his personal life.
“There is absolutely no skeleton, no information, scandal or otherwise that caused me to make this decision,” he told the Sunday Mirror.
“The reason I made the decision is because of the impact it has had on my own life and, principally, the lives of the loved ones around me.”
Asked if he feared a story involving drugs or his sexuality was about to emerge, Mr Umunna insisted: “No, no, no. When I’ve been asked about drugs I’ve been very open about that and other issues.
“Honestly, there is no revelation or no skeleton. You don’t run for the leadership of a major political party in this country if there is some information you don’t want to get into the public domain.”