Jeremy Corbyn has written to the prime minister, offering to throw Labour’s support behind her Brexit deal if she makes five legally binding commitments – including joining a customs union.

The Labour leader held private talks with Theresa May last week for the first time since her deal was rejected by a historic margin of 230 votes in January.

In a follow-up letter sent on Wednesday, he laid out in the clearest terms yet what commitments he is seeking in exchange for offering Labour support.

His intervention will dismay backbench Labour MPs and grassroots activists still hoping he will switch the party’s policy towards demanding a second Brexit referendum – which is not mentioned in the letter.

And it comes as No 10 prepares to publish legislation underpinning workers’ rights, perhaps as early as next week, in an attempt to win support from Labour backbenchers.

In his letter, Corbyn calls for the government to rework the political declaration setting the framework for Britain’s future relationship with the EU – and then enshrine these new negotiating objectives in UK law, so that a future Tory leader could not sweep them away after Brexit.

He says the changes to the political declaration must include:

A “permanent and comprehensive UK-wide customs union”, including a say in future trade deals.

Close alignment with the single market, underpinned by “shared institutions”.

“Dynamic alignment on rights and protections”, so that UK standards do not fall behind those of the EU.

Clear commitments on future UK participation in EU agencies and funding programmes.

Unambiguous agreements on future security arrangements, such as use of the European arrest warrant.

There is no mention of the second of the shadow Brexit secretary Keir Starmer’s six tests, which said that Labour would not support any deal that failed to deliver the “exact same benefits” as single market and customs union membership.

Labour’s approach, which Corbyn called “constructive”, appears to be focused firmly on the forward-looking political declaration, rather than the 585-page withdrawal agreement, which contains the Irish backstop and the divorce bill.

“We recognise that your priority is now to seek legally binding changes to the backstop arrangements contained within the withdrawal agreement, as we discussed when we met,” Corbyn writes.

“However, without changes to your negotiating red lines, we do not believe that simply seeking modifications to the existing backstop terms is a credible or sufficient response either to the scale of your defeat last month in parliament, or the need for a deal with the EU that can bring the country together and protect jobs,” he says.

He does not explicitly say that article 50 should be extended – a move that the shadow foreign secretary, Emily Thornberry, described as “sensible” on Wednesday.

But he said that because of the government’s negotiating failures “time has run out for the necessary preparation and for legislation to be finalised”. He added: “Following last week’s rejection by the House of Commons of no deal, all necessary steps must be taken to avoid such an outcome.”

He concluded: “My colleagues and I look forward to discussing these proposals with you further, in the constructive manner in which they are intended, with the aim of securing a sensible agreement that can win the support of parliament and bring the country together.”

May has so far vehemently resisted the prospect of reopening the political declaration, focusing instead on securing changes to the Irish backstop, which has been the flashpoint for Brexiter Conservative MPs.

But at the same time, her government has been trying to win over Labour backbenchers with new legally binding promises on workers’ rights.

She could propose a new draft bill as early as next week guaranteeing that UK workers’ rights will keep pace with Europe.

Tusk warns of 'special place in hell' for those who backed Brexit without a plan Read more

Backbench Labour sources said the new bill was in the late stages of drafting and would include a “regression lock” that the UK parliament would ensure workers’ rights would never slip below European standards. That approach appears likely to come close to meeting one of Corbyn’s five tests.

Multiple Labour MPs who have met May over the past few months have made it clear to her that they needed the guarantees to be made in primary legislation – because they do not trust that May will remain in No 10 for much longer.

Those who are in negotiations with the government have insisted that the bill must be tabled and have had at least one day of second reading within the next few months.

The Labour MP John Mann, one of the key figures behind the push for legislation, said: “There needs to be a bill produced soon. We aren’t basing anything on a promise, with all due respect to the prime minister. We are doing it based on statute.”

The Labour MP Lisa Nandy, who has also intimated she could back May’s deal, said the right offer from No 10 could win over up to 60 Labour MPs.

“If she were able to come back, and stop this eternal circular conversation within the Tory Party, and start reaching out to the rest of parliament and the rest of the country and give us those assurances, I think you’d get somewhere between 40 and 60 Labour [MPs] who would be prepared to step forward and say: we’ve got to come together around this, otherwise we leave with no deal,” Nandy told BBC Radio 5 Live.

But Phil Wilson, the Labour MP for Sedgefield and a leading supporter of the People’s Vote campaign, said: “It was perhaps inevitable that at some point my party would publish something like this even though most Labour MPs, members and voters have long since worked out that there is no form of Brexit that can meet the promises made in 2016 or do anything but make people poorer … In the end there is only one way out of this mess for both our country and the Labour party – a people’s vote.”

Trade unions are understood to have been pushing for the legislation to include an easing of restrictions on union activity. Union leaders had been seeking more concrete proposals for regulation of the labour market after Brexit and the enforcement of existing law.

“Their desire to bring in seasonal agricultural workers from around the globe, as well as barista visas, really does ring alarm bells,” one union source said. “Our fear is that when workers depend on their bosses for their visas, exploitation is sure to follow.”