Jeremy Corbyn’s shadow cabinet is set to clash again over Brexit this week, with supporters of a second referendum concerned that the Labour leadership will opt to facilitate a soft Brexit. With senior Labour figures openly calling for another public vote at the anti-Brexit march in London on Saturday, other influential MPs believe Corbyn’s inner circle is actually warming to a Norway-style Brexit that would see Britain leave the EU, but remain closely aligned to it.

Tensions between Labour and its pro-Remain activists are already high after the party released a tweet on Friday evening asking if supporters had any “big weekend plans” and called on them to go out leafleting for May’s local elections.

The party’s official position is to explore all possibilities to resolve the Brexit impasse, including a public vote if other avenues prove impossible. The position has allowed Labour to keep its options open amid widespread support for a second referendum among party members. However, a plan to hold a series of indicative Commons votes this week on possible Brexit options is set to force Labour to decide whether it can allow its MPs to back a soft Brexit.

Facebook Twitter Pinterest Labour’s deputy leader Tom Watson had a specific message for Theresa May during Saturday’s march. Photograph: Isabel Infantes/AFP/Getty Images

A previous Commons vote over a Norway-style soft Brexit last year resulted in a huge row within Labour. Jeremy Corbyn told his MPs to abstain, but 75 MPs voted for the idea, while 15 voted against. Six quit their Labour roles as a result of the vote.

But several Labour sources believed the party leadership was now warming to a Norway-style Brexit, dubbed “Common Market 2.0”, which would see Britain remain inside a customs union with the EU and part of its single market. Customs union membership would stop Britain signing its own trade deals, while single market membership would force Britain to adhere to the EU’s free movement rules on immigration.

Unlike the prime minister Labour's willing to be flexible to find a way through, but we won't vote for her terrible deal Rebecca Long-Bailey

Corbyn had seemed to be edging to the idea of backing a “confirmatory referendum”, which would see May’s deal put to the people. That idea has been drawn up by Labour backbenchers Peter Kyle and Phil Wilson, and will be voted on when May puts her Brexit deal to a third vote in the Commons, expected this week.

That option is still the preference of some in Corbyn’s team, said to include shadow Brexit secretary Keir Starmer. Tom Watson, the party’s deputy leader, appeared at Saturday’s march for a second vote and said he would back May’s deal if she put it to a public vote, with an option to Remain. “When the deal fails this week we are going to face the prospect of a huge delay to the process – I don’t think the people that voted Leave want that and I don’t think people who voted Remain want that,” he said. “So I have an explicit message for Theresa May.

“I will support your deal going through parliament or a revised deal you can agree with my party. I will help you get it over the line to prevent a disastrous no deal exit. But I can only vote for a deal if you let the people have a vote on it too.”

The rally was also addressed by London mayor Sadiq Khan, who said it was “time to withdraw article 50”. Speaking from the stage in Parliament Square, he said he was “a proud European”.

“No matter how you voted in the referendum, no matter what political party you support, we can all agree that Brexit has been a complete and utter mess,” he said. “With days to go we’re in danger of falling off the cliff, which will have catastrophic consequences... it’s time to give us, the British people, a final say on Brexit.”

Play Video 6:17 'People’s vote' march: up close with anti-Brexit protesters at the 'biggest ever demo' – video

Writing in the Observer on Sunday, shadow business secretary Rebecca Long-Bailey said parliament should now “consider the other options – including our alternative plan, the Common Market 2.0 proposal, a customs union, and a public vote”.

“Labour’s starting point is our alternative plan based around a new customs union and a close alignment to the single market. It abides by the result of the referendum without wrecking our economy. And it will work for the whole country, not just those at the top,” she said. “Unlike the prime minister, Labour is willing to be flexible to find a way through. But we will not vote for Theresa May’s terrible deal, and we’ll never countenance a disastrous no-deal crash.”

Some Labour MPs are also backing an amendment that would order May not to back a no-deal Brexit. It would give parliament the chance to block a no-deal outcome, a week before departure is set to take place.