In a letter sent to May as she meets EU leaders in Brussels on Thursday, Labour leader Jeremy Corbyn suggested her efforts to modify the deal itself to amend or remove the controversial "backstop" arrangement on the Irish border would fail.
But he said that if she secured changes to an accompanying political declaration on future ties before Britain leaves the EU on March 29, then his party could support the agreement.
The changes include a commitment to a "permanent and comprehensive" UK-EU customs union, in which Britain has a "say" on future EU trade deals.
Labour also wants "close alignment" with the EU's single market, underpinned by shared institutions and obligations. Corbyn said there should also be "dynamic alignment on rights and protections", commitments to maintain Britain's participation in EU agencies and funding programmes, and continued access to security databases and the European Arrest Warrant.
"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," he wrote.
Corbyn noted the EU's determination to keep the backstop, but said that a relationship along the lines he has suggested "would make it far less likely that any backstop arrangements would ever be needed".
Labour MPs who oppose Brexit responded with fury to the plan.
"Seriously? Offering to help Tory govt enable Brexit?" tweeted Chris Leslie.
"When the jobs go and revenues for services dry up as a result, Labour's leadership will have zero right to complain." Another, Chuka Umunna, added: "This is not opposition, it is the facilitation of a deal which will make this country poorer.
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