Labour’s plan for a permanent customs union with the EU after Brexit can secure a Commons majority, the shadow chancellor has told the BBC.
John McDonnell said it was a “very traditional British compromise” to avoid a “catastrophic” no-deal Brexit.
“We believe that this is a deal that could fly within Parliament,” he said.
Jeremy Corbyn’s letter setting out Labour’s demands for supporting a Brexit deal was welcomed by some EU figures and Tory MPs.
Downing Street is expected to reply to Mr Corbyn’s letter later on Friday.
In contrast to Mrs May’s deal, Labour wants the UK to be a member of a customs union with the EU, with an agreement “that includes a UK say on future EU trade deals” and close ties to the single market.
Under Mrs May’s plan, the UK would leave the customs union, which she says would allow it to strike trade deals around the world.
A senior No 10 source said the government was looking at Labour’s proposals “with interest” but added: “There are obviously very considerable points of difference that exist between us.
“The PM continues to believe an independent trade policy is one of the key advantages of Brexit.”
The UK is due to leave the EU on 29 March when the two-year limit on withdrawal negotiations under the Article 50 process expires.
But Mrs May has been unable to get the withdrawal deal she has negotiated with the EU through Parliament – it was overwhelmingly rejected by MPs last month.
In Brussels on Thursday, she told EU leaders that she could get a “stable majority in Parliament” for the deal if they agreed to legally-binding changes to the Irish backstop clause – something they have always ruled out.
Talks are continuing with EU officials – but senior figures in Brussels gave a warm reception to Mr Corbyn’s alternative proposals.
The European Parliament’s chief Brexit negotiator Guy Verhofstadt said: “It’s important now that this leads to a position in the UK that has the broadest possible majority, so that we can conclude these negotiations.”
European Council President Donald Tusk also described Mr Corbyn’s letter as a “promising way” out of the impasse, according to an EU source.
Cabinet Office Minister David Lidington has said he is willing to discuss the proposals with Labour’s shadow Brexit Secretary Sir Keir Starmer.
Mr McDonnell told BBC Radio 4’s Today programme: “The prime minister has to accept that the only way she will get something through Parliament is a compromise like this.
“I think if Theresa May said ‘I will sign up to Labour’s deal’ and we went to Parliament, I think we would have a secure Parliamentary majority.”
Conservative MP Sir Oliver Letwin was among those suggesting Mr Corbyn’s move could open the way to a cross-party consensus, if Mrs May could not get her deal through:
But Labour’s position has upset some of the party’s own backbenchers who see it as facilitating a “Tory Brexit” that they say will harm their constituents.
Some Labour members of the People’s Vote campaign for another EU referendum have accused Mr Corbyn of abandoning his commitment at Labour’s conference to get behind a public vote if he can’t force a general election.
Owen Smith, who failed in his bid to topple Mr Corbyn in a 2016 leadership vote, has said he and “lots of other people” were considering their future in the party as a result.
Asked about Labour opposition to Mr Corbyn’s offer, Mr McDonnell said “not everyone’s going to get everything they want” and MPs would have to compromise in the long-term interests of the country – but denied it had effectively killed off the prospect of Labour backing another referendum.
He said people had “looked over the edge of a no-deal Brexit” and economic growth was already stalling: “Therefore now in the national interest we have got to come together and secure a compromise. If we can’t do that, well yes, we have to go back to the people.”
Other Labour backbenchers have welcomed Mr Corbyn’s move. Labour’s Stephen Kinnock, who backs the “Norway Plus” model of a close economic partnership with the EU, tweeted: “This can break the deadlock.”