MPs on all sides of the Brexit divide have vowed to vote down Theresa May’s withdrawal agreement when the government brings the bill to the House of Commons next month.

Downing Street said the key piece of Brexit legislation – the withdrawal agreement bill – would be voted on in the week beginning 3 June. “It is imperative we do so then if the UK is to leave the EU before the summer parliamentary recess,” a spokesman said.

Talks with Labour are to continue in the meantime. The prime minister met the Labour leader, Jeremy Corbyn, on Tuesday night, though no substantive progress has been made towards a cross-party agreement.

The Democratic Unionist party’s Westminster leader, Nigel Dodds, said it was “highly likely” May’s deal would be defeated again unless the prime minister can “demonstrate something new that addresses the problem of the backstop”.

He added: “The prime minister has not pursued the one option that has ever achieved a positive vote for something in parliament. Alternative arrangements to the backstop won easily while everything else has failed.

“For the bill to have any prospect of success then there must be real change to protect the economic and constitutional integrity of the United Kingdom and deliver Brexit.”

The former environment secretary Owen Paterson, a member of the European Research Group, said nothing new had been put forward to convince hardline Brexiters in the Conservative party to vote for the agreement, and warned about the rise of Nigel Farage’s Brexit party.

“Sadly we will vote against it, yes, because as the DUP said in their statement, it doesn’t change the essential nature of the withdrawal agreement, which is unacceptable,” he told BBC Radio 4’s Today programme on Wednesday.

“People aren’t flocking to this new [Brexit] party’s rallies from the Tory party and the Labour party clamouring for the agreement, which has been the reaction of the government. They are going because they want to leave and they voted to leave, and they thought we were leaving on 29 March, and they feel thwarted and cheated.”

Remain-backing Labour and Liberal Democrat MPs also said they would vote down any agreement unless it had a referendum attached. The Lib Dem leader, Vince Cable, said his party would back any deal if it were subject to a fresh referendum.

“If the government were to put forward its deal tomorrow, however modified by the Labour party, we’ve made very clear we would support it, providing it is subject to a confirmatory referendum, so we are not in any sense blocking progress,” he told Today.

“I think now the government are running out of other options, I think that is where we are going to end up.”

Labour’s Anna McMorrin, a supporter of the People’s Vote campaign, said it was “yet another futile attempt to bounce parliament” and would fail again.

“They can try the same failed strategy as many times as they like, but the reality is there is no stable majority in parliament for their already thrice-rejected deal or for any other form of Brexit,” she said.

“The Labour party leadership must not be complicit in inflicting any botched version of Brexit on our country without letting the public have the final say.”

The Scottish National party’s Westminster leader, Ian Blackford, also said his 35 MPs would vote against the bill. “As is now beyond doubt, there is no such thing as a good Brexit. Any form of Brexit would destroy thousands of Scottish jobs and be catastrophic for living standards, our public services and the economy,” he said.

Bringing the withdrawal agreement bill to parliament with no stable majority is fraught with risk for the government. Should the bill fail to pass at second reading, May would not be able to bring it back during this parliament and may be faced with proroguing parliament in order to make any new attempt.