Theresa May’s Brexit bill is likely to pass through the Commons without major amendment next week, as Conservative rebels are backing away from supporting changes proposed by Labour or other opposition parties.

A band of Tory MPs fighting against a hard Brexit are indicating they have been largely satisfied by the prime minister’s promise of a white paper, which they believe could be published as early as Thursday.

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Labour and the Liberal Democrats now believe there is very little chance of getting enough cross-party votes for amendments. They had hoped to win support on issues such as guaranteeing the rights of EU nationals, and a more meaningful vote at the end of the two-year negotiations or protections in the House of Commons.

Opposition parties are now concentrating on getting the government to concede points voluntarily, with Labour MPs believing the most likely proposal to be accepted is a demand for May to provide quarterly updates to parliament on the process of negotiations.

MPs are due to start debating the bill in parliament on Tuesday. The legislation would give May the power to invoke article 50 and start two years of negotiations to leave the EU. The Commons will debate the bill for two days before a vote expected on Wednesday night. More detailed scrutiny involving proposed amendments to the legislation will begin next week.

The government was forced to bring legislation to parliament after the supreme court ruled that May did not have the power to trigger article 50 without the permission of MPs and peers.

Ahead of the first day of debate, May said MPs faced a “simple decision: do they support the will of the British people or not?”

“My message to people is very clear,” she said during a press conference in Ireland. “The people of the United Kingdom voted on the 23 June last year, they voted in a referendum that was given to them overwhelmingly by parliament. Six to one parliament voted. The people spoke in that vote, and the majority voted to leave the European Union, the job of the government is to put that into practice.”

One MP engaged in cross-party discussions said there was a reluctance of Tory MPs to step out of line, but there were some who were still thinking of ways to use this bill to nudge May away from the hardest Brexit path.

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However, a senior Lib Dem source said there was “no chance” of getting any substantial amendments passed with cross-party support and the debate was likely to be a “damp squib”.

If the bill proceeds without amendments, more Labour frontbenchers may have to resign rather than vote in favour of triggering article 50 as the bill stands. Dozens of Labour MPs are likely to vote against the bill but the legislation is likely to pass overwhelmingly with the support of almost all Conservatives and the Labour frontbench.

The biggest bloc of votes against article 50 on Wednesday is likely to come from the SNP, whose 54 MPs will oppose the legislation from the outset. Stephen Gethins, the SNP Europe spokesman, also criticised the prime minister for failing to publish the promised white paper that has left parliamentarians and the public heading blindly towards leaving the EU and policies based on “soundbites rather than sound arguments”.

There is a greater chance that the bill could be amended in the House of Lords, although peers are likely to avoid the appearance of trying to frustrate the bill because they are unelected parliamentarians.

The government announced on Monday that peers would debate the legislation after the February parliamentary recess, after it clears the House of Commons on either 8 or 9 of the month.

It will then be introduced for scrutiny by the Lords, where the government does not have a majority, on Monday 20 February, before completing its passage through the House of Lords probably on 7 March. If peers make any amendments, it would have to return to the House of Commons, where MPs would debate whether to keep the changes or get rid of them.

That would see the bill repeatedly move between the Commons and the Lords until an agreement were reached on the final text.

May is aiming to have the bill passed through both houses to meet her self-imposed deadline of triggering article 50 by the end of March.