Jeremy Corbyn is facing claims that Labour MPs and members could quit the party over his stance on Brexit.
The Labour leader has written to the Prime Minister setting out five demands that need to be met to get his party’s support for a Brexit deal.
But Owen Smith, his former leadership rival, said he was considering his position in the party and hit out at the prospect of Labour MPs being asked to back a deal which he said would damage livelihoods.
Other pro-EU MPs also criticised Mr Corbyn’s plan, claiming it was a change to the party’s policy of keeping open the prospect of a second referendum if Labour couldn’t force a general election.
Shadow Brexit secretary Sir Keir Starmer insisted the move by Mr Corbyn did not take the option of a so-called People’s Vote off the table.
In his letter to Mrs May, the Labour leader set out five demands:
A permanent customs union to deliver frictionless trade and help avoid a hard border on the island of Ireland
Close alignment with the single market, including shared institutions and obligations
Alignment on rights and protections so that UK standards keep pace with those across Europe as a minimum
Participation in EU agencies and funding programmes
Unambiguous agreements on the detail of future security arrangements, including access to the European Arrest Warrant and vital shared databases
On a visit to Worcester, Mr Corbyn told reporters: "Half of our trade is with Europe. A lot of our manufacturing industries are very frightened, very worried at the moment that on March 29 there’ll be a cliff edge.
"There cannot be a cliff edge. We will do everything we can in Parliament to prevent this cliff edge exit."
But Mr Smith, who stood against Mr Corbyn in 2016, said he was considering his future in the party.
Prominent pro-EU MP Chuka Umunna said the position was "totally demoralising".
He said: "This is not Opposition, it is the facilitation of a deal which will make this country poorer."
He added: "I hate to think what all those young voters who flocked to the party for the first time in 2017 will make of this.
"Vote Labour, get a Tory Brexit. They will feel they have been sold down the river."
Downing Street confirmed that Mr Corbyn’s letter had been received, and a spokeswoman added: "We will reply in due course. Our positions on many of the issues in the letter are well known. Our position on the backstop has not changed."
The spokeswoman said no further meeting between Mrs May and the Labour leader to discuss Brexit had been scheduled, but added "her door is open".
Mrs May’s effective deputy prime minister, David Lidington, said the Labour plan for a customs union with a say in EU trade deals was "wishful thinking".
In Brussels, a European Commission spokesman declined to discuss Mr Corbyn’s proposals, saying only: "Our interlocutor is Her Majesty’s Government and the Prime Minister."
On Wednesday European Commission president Donald Tusk highlighted Mr Corbyn’s "pro-Brexit stance" and said "there is no political force and no effective leadership for Remain".