Rebel MPs from both sides of the House of Commons are set to push for a Norway-style Brexit agreement which would keep the UK in the single market.
Both Prime Minister Theresa May and Labour leader Jeremy Corbyn have rejected such a deal, but pro-EU MPs claimed remaining in the European Economic Area (EEA) would be the best Brexit deal for jobs and businesses.
The issue will be voted on by MPs when the Brexit legislation returns to the House of Commons next week, with Labour divisions exposed by an amendment tabled by shadow cabinet ministers which stops short of backing EEA membership.
The official Labour amendments to the Brexit Bill call on Mrs May to make maintaining “full access” to the EU “internal market” an objective of the negotiations with Brussels.
Party critics have accused the frontbench of “bailing out” the Prime Minister from a potential rebellion on the single market.
With Mrs May lacking a majority in the Commons, she is vulnerable to any Tory revolt and at least 12 Conservatives have signalled their support for a future in the EEA.
But Sir Keir said the EEA amendment to the EU (Withdrawal) Bill that will go before MPs on June 12 was too divisive in the Parliamentary Labour Party (PLP).
He told BBC Radio 4’s Today programme: “The difficulty with that, and I think everybody recognises this, is that there are very strong and very different views across the PLP on that particular amendment.
“So, whilst there’s unity on all the others and we will all be voting together, on that amendment there are very divided views.”
Asked if he was putting party unity before his beliefs about what is right for the country, he replied: “I’m injecting some honesty about where we are in the Labour Party.”
But Labour MP Chuka Umunna said he would rebel to support the EEA plan – as well as the official frontbench amendment.
He said: “The option of us continuing to participate in the single market is only dead insofar as the Labour frontbench seek to adopt that position and I sincerely hope that we will do right by our values and seek to keep us in the EEA.”
At an event hosted by the Open Britain campaign, he told the Press Association he would rebel on Tuesday to support the EEA amendment and a “large number” of Labour colleagues would do the same.
“Every single MP needs to make their own decision about this in the national interest and in the interest of their own constituents,” he said.
“This is one of those issues where if you just blindly follow orders people will not forgive you.”
Former shadow chancellor Chris Leslie said: “We should not be in the business of bailing out Theresa May when she’s facing rebellion on the single market.”
Tory support for the EEA has been signalled in an amendment to the Trade Bill.
Former minister Stephen Hammond is thought to be one of at least 12 MPs behind it.
Pro-EU Tory Anna Soubry, who supports the EEA option, said a majority of MPs would back remaining in the single market if it was put to a free vote but a “culture of fear” was preventing them from speaking out.
At the Open Britain event, she said that Tories at all levels of government – including in the Cabinet – supported an EEA deal.
“They will not speak out, they will not be true to what they believe in.
“That has got to change, we cannot allow this culture of fear that has developed to continue any longer.”
Former minister Ms Soubry added that people should be “angry” that debate on the Lords amendments to the EU Bill was being limited to a single day and she was scathing about the Government’s handling of the Brexit process.
“We have a Cabinet that is completely and totally divided so there is no plan for what the negotiating position is going to be,” she said.
“You may think that is a pretty shabby and shocking state of affairs. That is the reality.”
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.”
Brexit Secretary David Davis is set to make a speech on his hopes for the future security partnership between the UK and the EU.
The intervention comes after a bruising row over the Galileo satellite programme and European arrest warrant which saw Mr Davis suggest Brussels was concerned with “public posturing and scoring points” rather than resolving the issues.