Jeremy Corbyn will be the “midwife of a hard Brexit” that will harm Britain’s poorest unless he fights to stay in the EU single market, David Miliband has warned.

The former Labour foreign secretary – launching a new cross-party campaign to soften the exit terms – said his own party must shift its stance, or punish its own natural supporters.

The crucial decision about whether the UK stays in the single market is now effectively in Mr Corbyn’s hands, ahead of a crunch Commons vote – prompting Mr Miliband to urge him to give his backing.

“The warning for Jeremy Corbyn is that, if he is not very careful, he will be the midwife of a hard Brexit that threatens the living standards of the very people that he says he wants to represent,” he said.

The single market was about workers’ rights, as well as trade, he argued, warning: “This is an issue for MPs and leaders of all parties, because the stakes could not be higher. This goes beyond party politics – it’s more important than party politics.”

The House of Lords voted to make keeping the UK inside the European Economic Area (EEA), and therefore the single market, a negotiating priority – throwing the decision back to MPs.

But Labour remains opposed, while vowing to negotiate a “strong single market relationship that hardwires the benefits”, without explaining how that will be possible outside the trading bloc.

Mr Miliband was speaking ahead of sharing a platform with Nicky Morgan, the former Conservative education secretary, and Nick Clegg, the former Liberal Democrat deputy prime minister.

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The trio are warning that Britain is being “held to ransom” by hardline Tories hell-bent on a hard Brexit, in the starkest evidence yet that party allegiances are breaking down as battle rages over departure from the EU.

Speaking to BBC Radio 4’s Today programme, Mr Miliband dismissed chatter about a new centrist party, saying his campaign was pushing for “cross-party working, not for creating a new party”.

Bitter cabinet splits made it a “national crisis”, he warned, adding: “The UK has no negotiating position on the fundamental issues that face the country as it contemplates its post-Brexit relationship with the European Union.”

He predicted an “enormous sigh of relief in Europe” if Britain stayed in the single market, because the EU “don’t want a weakened Britain, because they think it will weaken Europe”.

The “Norway option” was now “on the table”, after that country’s prime minister made clear it would welcome the UK staying in the EEA.

Mr Miliband, who runs a UN charity in New York, said he was also seeing at close quarters how the US, under Donald Trump, was turning away from Europe. “We have to take that into account,” he said.

And he rejected Brexiteer protests that revolts in parliament weakened Britain’s negotiating hand, arguing: “What’s more destructive for a government position than having no position?”

The three prominent politicians are calling on MPs to stand up to “the hardest of Brexits”, which they characterise as “a fantasy island” of “economic pain” and diminished standing.

“Over the coming months, MPs will have the chance to table amendments to bills – and vote for those amendments – which can prevent the country from suffering the long-term damage that a hard Brexit will cause,” they wrote at the weekend.