Theresa May summoned several pro-leave Tory MPs to Chequers on Friday, just after president Trump’s departure, as she embarked on a charm offensive aimed at avoiding embarrassing defeats in two key Brexit bills next week.

With a vocal section of the Conservative party in open revolt over Thursday’s white paper on the future relationship with the European Union, the prime minister is holding private meetings with backbenchers to try to win their support.

One senior Brexiter described it as “an effort to butter up the butterable”, claiming that a number of hardliners had turned down May’s invitation.

Two crucial pieces of Brexit legislation – the customs and trade bills – make their long-delayed return to the House of Commons on Monday and Tuesday.

Tory remainers have laid amendments aimed at keeping Britain in a customs union. But with the debate in cabinet moving in the direction of a somewhat softer Brexit, after the resignations of David Davis and Boris Johnson, all but the most outspoken are now wary of further destabilising the prime minister.

Instead, May’s most pressing challenge is likely to be heading off a backlash from members of the well-drilled European Research Group of Tory backbenchers, chaired by ardent leaver Jacob Rees-Mogg, which has tabled a series of amendments. These are unlikely to pass because they will not get Labour support, but the government could face an embarrassing defeat if more than a handful of hardline Brexiters abstain from backing either bill.

Labour voted against the second reading of the bills the last time they came before MPs and a senior party source said,: “We are treating both days very seriously”.

The leader of the House of Commons, Andrea Leadsom, confirmed on Thursday that the two bills would be debated at the start of next week – though some at Westminster believed she was preparing to delay the discussions until after the summer break at the request of the chief whip.

Davis and Johnson have so far been absent from the chamber of the House of Commons since they resigned over the Chequers compromise that May hopes will finally allow negotiations with Brussels to make progress.

Both are expected to make a reappearance on Monday, with Downing Street watching Johnson’s moves, in particular, very carefully.

No 10 felt May survived the spate of resignations with her authority largely intact and took the opportunity to reshuffle her cabinet, promoting remainers Jeremy Hunt and Matt Hancock. Her aides now hope she can survive until parliament’s long summer recess without too many more crises.

A vote of no confidence in May would be triggered if 48 Conservative MPs wrote letters to Sir Graham Brady, who chairs the backbench 1922 committee – though many of her backers believe she could face down such a challenge. Under party rules, she would then be safe from another confidence vote for 12 months.

The ERG was infuriated by the white paper. The 98-page document sets out plans for a “common rule book” for goods and food, alongside a looser trading relationship for services. It would also see high-skilled workers and students from the EU allowed to continue to come to the UK to live and work and proposes a complex “facilitated customs arrangement”, which would see the UK collect tariffs on the EU’s behalf for some imports.

Rees-Mogg has claimed May’s approach will reduce Britain to “vassalage”, echoing Johnson’s claim that such a deal would reduce the UK to “the status of a colony”.

On Friday, Davis’s chief of staff, Stewart Jackson, who has claimed his reappointment to serve the new Brexit secretary, Dominic Raab, was blocked by Downing Street, compared May’s Brexit stance to the historic error of joining the European exchange rate mechanism (ERM).

Britain crashed out of the ERM on Black Wednesday, in October 1992, after trying and failing to halt a decline in the pound, shattering the Tory party’s reputation for economic management.

“Chequers could be another ERM. Voters are smart and have long memories. Such a fate can obviously be avoided,” he tweeted.

For the time being, the ERG is not openly advocating a change of Tory leadership – instead using what one Brexiter called “Spanish practices” to try to force the government to change course.

Davis’s number two at the Department for Exiting the EU, Steve Baker, a former ERG chair, also walked out in protest at the Chequers plan and has made clear he intends to use every parliamentary tactic available to oppose it.

Several other more junior Conservative figures also resigned earlier in the week, including two party vice chairs, Ben Bradley and Maria Caulfield. Bradley, who won the marginal seat of Mansfield in last year’s general election, said: “If we do not deliver Brexit in spirit as well as in name, then we are handing Jeremy Corbyn the keys to No 10.”

MPs who accepted May’s invitation to her country retreat on Friday included the Weston-super-Mare MP John Penrose and the veteran MP for Gainsborough, Sir Edward Leigh.