Labour must keep open the option of a second EU referendum with remain on the ballot paper, Keir Starmer has said.

After Theresa May’s Brexit deal was overwhelmingly voted down on Tuesday, leaving Labour as potential kingmakers in shaping the nature of Britain’s exit from the European Union, the shadow Brexit secretary said Labour must either instruct the government to negotiate a close economic relationship with the EU based on a customs union, or campaign for a further public vote.

It comes after a leaked poll commissioned by the pro-EU Best for Britain campaign suggested that voters would be less likely to back Labour if the party was committed to stopping Brexit, while a number of junior shadow ministers have said they would consider their positions if Corbyn conceded to pressure to back a second referendum.

Speaking at a Fabian Society event in London called Brexit and Beyond, Starmer restated the commitment made at the Labour conference in Liverpool last year, when the party pledged to keep all options on the table, including campaigning for a public vote – in the event that the opposition cannot force a general election.

“That is a very important commitment,” he said. “It’s a commitment to you, our members and our movement. And it is one we will keep.

“As I set out in Liverpool, a public vote has to be an option for Labour. After all, deeply embedded in our values are internationalism, collaboration and cooperation with our European partners.”

He reiterated on Saturday: “I don’t think it is any secret I firmly believe there should be a remain option – and there has to be a genuine leave option.”

The deputy Labour leader, Tom Watson, who also spoke at Saturday’s conference, called on his party not to fail Britain at a “great moment of change” and said that “the country needs the leadership that only we can give. Let’s make sure we do not fail them”.

The shadow culture, media and sport minister said that a second referendum might be required to break the deadlock in parliament. “That might be the only way that the insurance policy part of our conference resolution can help break the impasse in those Brexit negotiations,” he said.

Elsewhere, the Conservative MP Sarah Wollaston has said she will table a “people’s vote” amendment to the government’s plan B Brexit motion after it is published on Monday.

During his speech Starmer said it now appeared inevitable that the government would have to apply for an extension to the article 50 withdrawal process since there is a raft of legislation that must get through parliament before Brexit, with less than 70 days remaining before the UK’s slated exit date on 29 March.

“It seems inevitable to me that the government will have to apply for an extension of article 50,” he said. “And of course neither of the alternative options I have set out today could realistically be completed by 29 March.

“So, it’s time for us inject some honesty into this debate, and to identify the credible solutions that remain. We also have to recognise that nobody – whether you voted leave or remain – wanted to be in this position.

“In the coming weeks parliament will have the chance to take control. That starts by being open about the dilemmas we face, and the credible choices that are still available,” he added.

The former head of the Crown Prosecution Service also criticised the failure in the Brexit debate to “tackle the causes of the referendum result”. He stressed the need to “tackle inequality, low pay, a broken housing market and the growing dislocation between our political system and the people who elect us”.