David Cameron came under new pressure over Europe on Saturday as the frontrunner for the Labour leadership, Andy Burnham, called on him to bring forward his planned in/out referendum to next year and demanded a far-reaching renegotiation to address public concern over immigration.

The move means that, if Burnham is chosen by his party to succeed Ed Miliband in September, Labour will not only perform a dramatic U-turn by throwing its backing behind an in/out vote, but will also join Tory Eurosceptics in demanding a tough new settlement for the British people as soon as possible.

After the Conservatives’ election win, senior Tories also broke cover to warn the prime minister not to try to foist a minimal renegotiation on the public, saying they would not be fooled and that a poor deal would benefit Ukip and make the UK’s exit more likely.

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In an interview with the Observer, Burnham said he would overturn Labour’s policy of only holding a referendum if there were a substantial transfer of power to Brussels. He said he would back a referendum wholeheartedly, hoping for a new deal and a yes vote. He said the date should be moved for the sake of British businesses who had complained of continued uncertainty.

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.”

The former Labour health secretary, who is attracting strong backing within the party, said it was vital to end uncertainty about Britain’s place in the EU as soon as possible, and to use renegotiations to tighten rules on EU migrants claiming benefits after entering the country.

Burnham said he would back Cameron if he secured a good deal and brought forward the referendum – but would hold him to account if he tried to sell an unambitious settlement as more far-reaching than it was.

“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.”

His minimum requirement to ensure Labour support for a deal would be that there should be a two-year ban on citizens from other EU countries claiming benefits after they arrive in the country and a clampdown on the undercutting of pay in the UK by European migrants.

Cameron, who will embark on a tour of European capitals in the next few weeks as he begins the renegotiation effort, pledged before the election to hold the referendum by the end of 2017, but Downing Street has suggested it could be brought forward to next year.

The PM believes that he now has a mandate for reforms, including a ban on EU migrants from claiming any benefits, including tax credits, for four years. But opposition to such a move is sure to be mounted by many EU leaders, including those of eastern European nations whose citizens settle in this country and find jobs here in large numbers.

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Burnham said he was a “passionate pro-European”, but argued that if Cameron played his cards well it was possible for him to deliver substantial reform that would help the business community and ease concerns in working communities about immigration.

Meanwhile, Tory Eurosceptics, although facing calls from party loyalists to give Cameron a period of grace following the Tories’ election success, were already breaking cover, increasing pressure on him to take on EU leaders and set the bar high in talks.

Their concerns that he might be willing to accept a limited renegotiation, and try to win a yes vote with an early poll next year, were aroused by comments last week by foreign secretary Philip Hammond who suggested that the changes required on free movement of labour and benefits could be achieved without any changes to EU treaties.

Former Tory Europe minister David Davis told the Observer: “The government has to calculate very carefully and think that taking a minimalist approach will actually feed Ukip rather than starve Ukip in a referendum, and may alter the odds against the government winning a referendum and have implications for the next election.

“To minimise the expectation to a sub-optimal method is not a good tactic. In any negotiation the standard strategy is to ask for a little bit more than you might get, not quite a bit less than they might get.”

He suggested that the public would not be fooled by a fudged, limited deal. “It is never wise for a government to underestimate the intelligence of the British public.”

On Saturday another candidate in the race for the Labour leadership, Liz Kendall, also threw her support behind the in/out referendum. The issue was a divisive one in the party throughout Ed Miliband’s leadership, with key figures including policy chief Jon Cruddas arguing that Labour should back the referendum and allow the people to decide.

In an interview for this newspaper, Cruddas, who helped write the Labour manifesto, says the party is now facing arguably the biggest crisis in its history, and calls for an honest and far-reaching reassessment of its future.

Burnham, who will address the CBI this week in an attempt to display his pro-business credentials, accepted that the policy of embracing the referendum and setting the bar high for success was risky in that the result could be a no vote, but he believed it could and would be a yes.

“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 himserlf and he has to deliver.”