Jeremy Corbyn was under pressure from across the labour movement last night to support a softer form of Brexit as the TUC joined senior party figures in calling for Britain to retain the option of staying in the single market.

The unions’ intervention followed a call from Hilary Benn, the Labour chair of the Brexit select committee, who wants the party leadership to shift position and support staying in the customs union rather than rejecting the possibility outright.

Last week, 50 senior Labour MPs, peers and MEPs, including the former shadow business secretary Chuka Umunna, signed a statement saying the best way to respect the wishes of the pro-Remainers who voted for Labour on 8 June was to “fight unambiguously for membership of the single market”.

Both Corbyn and the shadow chancellor John McDonnell have thus far stopped short of explicitly insisting upon membership of the single market and customs union, stressing instead that whatever deal is done must ensure that the UK continues to enjoy the full benefits of both.

However, after the general election, there is a growing view in Labour, and sections of the Tory party, that the government lacks a mandate for a hard Brexit, and that such as an approach is not backed by a majority of MPs, who fear dire economic consequences of such a complete break.

Frances O’Grady, the general secretary of the TUC ,is known to back single market membership. She tweeted earlier this year: “If we leave the single market, working people will end up paying the price. It’d be bad for jobs, for work rights & for our living standards.”

On Saturday, a statement from the TUC said that Britain “needs a Brexit that works for working people” and insisted it was a “mistake” for Theresa May to rule out membership of the single market.

“That risks people’s jobs and their hard-won rights at work. At the very least, the Brexit deal must allow tariff-free and barrier-free trade with Europe. And written into the deal there must be an ongoing level playing field on workers’ rights between the UK and the EU.”

Labour MPs expect that unions such as the GMB may also pressure Corbyn to shift ground in the run up to the party conference in the autumn. With bills on Brexit dominating last week’s Queen’s speech, the battle over the terms of the UK withdrawal from the EU is about to begin in earnest in parliament.

Increasingly, pro-EU MPs in both Labour and Tory ranks are urging their leaders to modify their positions to ensure the economic damage is as limited as possible.

Speaking on BBC Radio 4’s Today programme Benn said maintaining tariff-free trade was essential to the future of the British economy and urged Corbyn to move. “The simplest way to do that is to remain in the customs union,” he said.

From the Tory side, the former minister Nicky Morgan – who is campaigning to become the chairwoman of the powerful Treasury select committee – said it was “very disappointing” that May had decided to “turn her back completely on the single market and the customs union”.

She said there was a growing view across parliament that membership of the customs union, at least, had to be kept on the table in Brexit negotiations.

An early test of cross-party cooperation among those MPs who are opposed to a hard Brexit will come this week when the Liberal Democrats put down an amendment to the Queen’s speech calling for Britain to stay in both the single market and customs union.

The Lib Dems say they have opened talks with MPs from other parties. Alistair Carmichael, the party’s chief whip, said: “Many people, including those who voted Leave, are increasingly worried about what Brexit will mean for our economy and living standards.

“We’re now staring over the precipice at the economic reality of an extreme Brexit. Future generations won’t forgive us if we fail to act.”

On Sunday the government will pledge to keep duty-free access to UK markets for the world’s poorest 48 countries across the globe, including Bangladesh, Haiti, Ethiopia and Sierra Leone. They will continue to benefit from duty-free exports into the UK for all goods other than weapons, known as the “everything but arms” EU initiative.