Soft-Brexit Tory MPs and independents are resisting the idea of attaching a customs union to Boris Johnson’s withdrawal agreement bill, meaning it would likely need an alliance of all opposition parties to pass.

Conservative MPs and former Tories, who previously backed the idea of a customs union, suggested on Tuesday they would prefer not to reopen the issue and wanted to support Johnson’s deal.

Damian Green, the leader of the One Nation caucus, and Oliver Letwin, who has is heavily involved in the parliamentary fight against a no-deal Brexit, both indicated they would back Johnson’s deal without a customs union appended.

However, Johnson is still facing the prospect that Labour could ally with enough opposition and independent MPs to amend his bill in favour of a softer Brexit.

Labour said on Sunday it would try to win support from the Democratic Unionist party and other opposition MPs for attaching a customs union to Johnson’s Brexit legislation. The DUP has not ruled out backing such an arrangement, although one of its 10 MPs, Jim Shannon, said it would not look favourably on a custom union.

The Lib Dems, who want to revoke article 50, and the SNP, who are pushing for an election, will also not say publicly how they would vote on trying to soften Johnson’s Brexit.

To pass, a customs union amendment would also need support from pro-deal Labour MPs. Gloria De Piero, a Labour backbencher who signed a letter to the EU urging it to work day and night to reach a deal, said she would work towards such an outcome.

Another pro-deal Labour MP told the Guardian he would back a customs union regardless of whether the amendment could halt the legislation in its tracks.

However, the government is likely to frame any vote on a customs union as a matter of confidence in Johnson’s Brexit bill. If such an amendment passed, the government would be likely to cancel the legislation and the prime minister could try again to force an election.

No 10 refused to comment on whether Johnson would pull the whole bill if a customs union were attached. But the prime minister’s official spokesman said: “Essentially if the legislation in the House of Commons steps too far away from what was agreed in the withdrawal agreement and the political declaration that does bring into question ratification.”

Labour is likely to seize on comments by Steve Barclay, the Brexit secretary, in making its argument for a UK-wide customs union, after he was forced to admit the current deal would require businesses to complete export declarations for goods moving from Northern Ireland to the rest of the UK.

Barclay originally told a Lords EU committee that it would not be necessary for businesses to make declarations. But five minutes later, he had to admit: “Exit summary declarations will be required in terms of NI to GB.”

Soft-Brexit Tories are understood to be more open to amending the withdrawal agreement bill to make sure parliament would have the power to stop a crash-out on World Trade Organization terms at the end of 2020. The UK would have to have struck a trade deal by this point or agreed an extension to the transitional period in order to avoid that.

Asked if no deal was still a possible scenario at the end of the transition period, Barclay said it was “not a credible scenario, because firstly we have just seen the prime minister deliver a deal when he was accused of having a sham negotiation and going for no deal.

“Secondly, both sides have agreed a framework for the negotiations and legal commitments to expedite those discussions and to act in good faith,” he said.

Soft-Brexit Tories and pro-deal Labour MPs are also keen to have parliamentary approval for the terms of the UK’s trading relationship with the EU, so one option for Johnson is to offer a vote on this in future.

The withdrawal agreement bill is expected to be published on Monday. MPs will then get a second reading vote on Tuesday about whether it should pass to its next stage in the Commons. Labour, the SNP, Lib Dems, DUP, Plaid Cymru and the Greens are all planning to vote against the bill, but there will be some Labour rebels and independents backing the Conservatives.

At that point, MPs will also have to vote on whether to accept the government’s accelerated timetable for the bill, allowing less than a week for its scrutiny. There is more chance that the government could be defeated on this point, forcing Johnson to take longer to pass the legislation and forcing an extension to article 50 beyond 31 October.

Johnson is expected to have enough support for the bill to pass its second reading but the tricky point for the prime minister is when the legislation reaches its committee stage, when MPs can table and vote on amendments.

If the bill gets through this phase unamended or with amendments acceptable to the government, it would then have to pass a final third reading of the bill – the crucial moment that will determine whether the Commons agrees to the overall terms of the legislation.