Cabinet members are not expecting to discuss Brexit at their scheduled meeting this week, despite the looming deadline of the European council meeting next week.

Brexit does not feature on copies of the agenda for the meeting on Tuesday, even as British officials rush to finalise the UK’s negotiating position in time for the next summit on 17-18 October.

Instead, the papers indicate the cabinet discussion will be dominated by a presentation on the budget by the chancellor, Philip Hammond, which is due at the end of the month.

Ministers expect to discuss Brexit in a week’s time when some hope that Downing Street will have clarified how the UK proposes to handle cross-border regulatory checks if no progress is made on agreeing a free trade deal with the EU.

There has been speculation that this could involve the whole of the UK agreeing to be part of a common customs area with the EU in order to avoid the possibility of a hard border separating Northern Ireland from Great Britain, in the event that no long-term trade deal is signed.

The emerging detail will be scrutinised by all sides, but it is unclear whether Tory hard Brexiters would accept such a backstop if it runs on for a long period, and in particular beyond the next general election scheduled for 2022.

Downing Street would not speculate further on timing, saying merely that its regulatory backstop proposals would emerge “soon”. They are likely to form the basis of technical negotiations with Brussels as officials scramble to find a form of words for the withdrawal agreement that the UK proposes to sign with the EU.

Parliament returns on Tuesday, and Tory backbenchers opposed to Theresa May’s Chequers proposals – including the pro-hard Brexit European Research Group (ERG) – are understood to be seeking an opportunity to stage a show of strength by forcing a vote in parliament, possibly doing so on an obscure measure such as a statutory instrument.

The numerical strength of the ERG has never been tested in a Commons vote. Tory whips have suggested there could be as many as 80 Brexit rebels – although that figure is considered by most observers to be high, ERG insiders insisted there could be more.

A total of 34 MPs have declared that they would not vote through a Chequers-based deal according to the Stand Up 4 Brexit campaign website, which would be enough to prevent May getting her deal through parliament if all the opposition parties also voted against it.

Failing that, some members of the ERG suggested they could vote down parts of the budget, which is due on 29 October, although the tactic would be particularly controversial given that the confidence of the government depends in part on its ability to get finance bills through parliament.

It emerged over the weekend that Downing Street is sufficiently worried about the prospects of squeezing May’s final deal through parliament that it has secretly begun drawing up plans to intensively canvass Labour MPs to see if they would support it.

The idea is that May and her principal allies would argue that her deal needed to be supported in the “national interest” because the only alternative on offer would be a no deal. A “Canada-style” deal favoured by the hard Brexiters would not, No 10 has repeatedly said, solve the problem of ensuring there was no disruption at the Irish border.

But both Labour and hard Brexit Tory MPs reacted badly to the idea that it might be possible to get May’s deal through with a bloc of support from the opposition. Only a tiny handful of pro-Brexit Labour MPs such as Kate Hoey are thought to be prepared to support May.

Alison McGovern, a backbench Labour MP who supports a second referendum, said: “If it were very possible to find a cross-party compromise then this approach has come too late. Keir Starmer, the shadow Brexit secretary, has set out our six tests so if the government wanted Labour votes they should have started there.”

Starmer’s six tests includes one that will ask of May’s final deal: “Does it deliver the ‘exact same benefits’ as we currently have as members of the single market and customs union?” while earlier this month Labour’s leader, Jeremy Corbyn, said his party would only support a deal that protected jobs, workers’ rights and maintained environmental and consumer standards.

Hard Brexit-supporting Conservative MPs told their WhatsApp group that if May somehow managed to get a Chequers-based deal through the Commons by relying on Labour votes, then they would be prepared to vote down future government legislation, meaning that she could not stay on as prime minister.

One of their number, Bernard Jenkin, told the WhatsApp group they should “Have no doubt” that “a soft/non Brexit pushed by the Conservative establishment etc, but put through with Labour support will look like we are abandoning our supporters and remove any sense of obligation amongst Conservative Brexit supporting MPs to continue to support the government”.

A No 10 source said: “We’ve always said we are working hard to get a deal this autumn. There are big issues to work through. And as the prime minister has said, this will require movement on the EU side.”

Other items up for discussion in Tuesday’s cabinet include an update on the race disparity audit one year on and ministerial activity for World Mental Health day on Wednesday.