Theresa May faced a growing Tory revolt over her leadership as it emerged that Remain-supporting Conservative MPs are being told by party whips that they will be seen as “supporting Jeremy Corbyn” if they attempt to soften the Brexit bill.

Furious Conservatives, including former ministers, said such threats and arm-twisting from the whips’ office would “backfire” spectacularly, making it more likely the prime minister would face a leadership challenge this autumn.

Tory tensions over Brexit, coupled with dismay over May’s insistence last week that she wants to lead the party into a 2022 general election, reached new heights as MPs prepared to debate the European Union (withdrawal) bill when parliament returns after the summer break on Tuesday.

One former cabinet minister told the Observer that May’s determination to cling on to office, and her hardline position on Brexit, would “definitely” trigger letters to Graham Brady, chairman of the backbench 1922 committee, over coming weeks. If 15% of Tory MPs write to Brady expressing no confidence in her, a leadership contest has to be called.

Another former minister, the outspoken pro-Remain MP Anna Soubry, told the Observer that it was unacceptable for whips to try to stop Conservative MPs tabling and backing amendments to the Brexit bill by threatening them.

“Any suggestion that this is any way treacherous or supporting Jeremy Corbyn is outrageous,” she said. “It amounts to a trouncing of democracy and people will not accept it.”

Several other Tories said the whips were on full manoeuvres, telling MPs that tabling amendments or supporting them would be tantamount to helping Corbyn into Downing Street.

Soubry said she saw it as her duty to do what she believed was in the national interest, rather than bow to intimidation and bullying by party managers. “People will be very alarmed. It will all backfire on them.”

Another senior Tory MP said: “If there is any more of this kind of nonsense from the prime minister or the whips then the feeling against her – which is already considerable – will grow to the point where her position is unsustainable.”

While the government is not expected to face serious trouble over the Brexit bill at the second reading, Tory MPs are already planning to table amendments at the committee stage in October, which could attract cross-party backing.

The most serious chance of a cross-party alliance forming is over amendments calling for the option of the UK remaining in the single market and customs union to be kept open, broadly in line with Labour’s new policy revealed in the Observer last weekend.

In a statement aimed at her own MPs, May makes clear that while they can debate and scrutinise the bill, they should back it because it expresses the will of the British people.

In what some will see as a veiled threat, May says: “Now it is time for parliament to play its part. The repeal bill delivers the result of the referendum by ending the direct role of the EU in UK law, but it is also the single most important step we can take to prevent a cliff edge for people and businesses, because it transfers laws and provides legal continuity.

“We have made time for proper parliamentary scrutiny of Brexit legislation and welcome the contributions of MPs from across the House. But for us to grasp the great prize ahead of us, that contribution must fit with our shared aim: to help Britain make a success of Brexit and become that great global country we know we can be.”

Labour will register its opposition to several clauses in the bill as it seeks to keep options open on the single market and protect workers’ rights.

A serious row is also brewing over the amount of time allocated to debate the bill – the so-called timetable motion – with Labour and Conservative MPs fearing it will be curtailed.

Labour MP Chuka Umunna, co-chair of the all-party parliamentary group on EU relations, said: “The House of Commons spent over 10 days and 20 days respectively in committee debating joining the EU in the 1970s and the Maastricht treaty in the 1990s.”.

“Government attempts to sabotage time to debate the repeal bill and how we withdraw from the EU should be fought tooth and nail. There should be no curtailing of the proper democratic process on issues of such critical importance.”

While May is under attack from many in her own party, a special poll on leadership by BritainThinks, published in the Observer, finds that far more people believe the prime minister will get the best deal for Britain out of Brexit negotiations (46%) than think Corbyn would (25%).

However, in a wide-ranging examination of the qualities that people now regard as most important in their leaders, there are signs that people increasingly value those associated with Corbyn’s success. The quality most people now regard to be most important in a leader is “integrity” – up 10% since the same questions were put two years ago – with “authenticity” up 5%.