Jeremy Corbyn is demanding a "new single market" deal for the UK after Brexit and he wants to make full access to the EU market a main objective.
But the move by the Labour leader to attempt to avoid a Labour split on Brexit in the Commons next week appears to have backfired already.
Last month in the House of Lords 83 Labour peers defied his order to abstain and voted for UK membership of the European Economic Area (EEA) - the so-called 'Norway model' - after Brexit.
The result was a government defeat, one of 15 in the Lords on the EU (Withdrawal) Bill, which the government will attempt to overturn in a marathon 12-hour Commons debate next Tuesday.
But instead of backing the Lords amendment on the Norway model, which is backed by pro-Remain Labour MPs, Mr Corbyn has come up with his own alternative proposal.
Labour is demanding a "new single market" deal for the UK after Brexit and calling on Theresa May to make maintaining "full access" to the EU internal market an objective in Brexit negotiations.
Pro-Remain Labour MPs have reacted furiously to Mr Corbyn's move, accusing him of a "fudge" and claiming he is poised to come to the Prime Minister's rescue when she is facing a rebellion by pro-EU Conservative MPs.
Explaining his proposal after addressing the GMB union conference in Brighton, Mr Corbyn said: "We are confident we can build a new relationship with the EU. We want the UK to have a better deal than the Norway model."
And Labour's shadow Brexit secretary Sir Keir Starmer said: "Labour will only accept a Brexit deal that delivers the benefits of the Single Market and protects jobs and living standards.
"Unlike the Tories, Labour will not sacrifice jobs and the economy in the pursuit of a reckless and extreme interpretation of the referendum result.
"Existing Single Market agreements that the EU has negotiated with third countries, including Norway, are bespoke deals negotiated with the EU to serve the best interests of those countries.
"We need to learn from them and negotiate our own more ambitious agreement, which serves our economic interests and which prevents a hard border in Northern Ireland.
"Two years on from the referendum it is clear that the government has no plan for how it will protect jobs and the economy, and guarantee no hard border in Northern Ireland.
"Labour's amendment, along with a commitment to negotiate a new comprehensive customs union with the EU, is a strong and balanced package that would retain the benefits of the Single Market.
"Parliament should have the opportunity to debate and vote on it."
But leading a pro-Remain backlash, Labour MP Chris Leslie said: "Labour members up and down the country will be aghast that Jeremy Corbyn is going to bail out Theresa May next week, just as we have Tory rebels ready to join us on EEA."
And Labour MP Chuka Umunna, a supporter of the pro-EU Open Britain campaign, said Mr Corbyn's proposal was "a fudge" and "nowhere near enough".
He said: "All the way through the passage of this Bill, the only amendments which have commanded support on both sides of the House and passed are cross-party backbench ones.
"So, if we are serious about 'protecting full access to the internal market of the EU' and ensuring 'no new impediments to trade', logic dictates Labour MPs should be whipped to support the cross-party EEA amendment sent to us by the House of Lords."
Conservative junior Brexit minister Suella Braverman said: "Labour have shattered their promise to respect the referendum result.
"This amendment means accepting free movement and continuing to follow EU rules with absolutely no say in them, which is the worst of all worlds.
"Only the Conservatives will get the best deal for the whole country - delivering on the referendum vote to get control of our money, borders and laws, while building a strong new relationship with Europe."
The Lib Dems accused Labour of "peddling snake oil in a bottle wrapped with the EU" with their amendments, insisting the only way to access the Single Market is to be part of the EEA.
The Labour Party's move comes as a new opinion poll suggests a wobble in public support for Brexit, nearly two years after the EU referendum in June 2016.
Asked by pollsters YouGov, "In hindsight, do you think Britain was right or wrong to leave the EU?", only 40% said it was right and 47% said it was wrong.