Rory Stewart has declared he wants the prime minister’s job, becoming the first cabinet minister to openly do so, as he warned the Conservatives the party would lose 4m votes if it pursued a hard version of Brexit to “outdo Nigel Farage”.

Stewart, who was made international development secretary this week, said “yes” when asked if he would throw his hat into the ring to become prime minister, but insisted he was not keen for Theresa May to step down immediately.

It comes before a crucial week for May when she is likely to be told by the 1922 Committee that she must set a fast timetable for her departure or face another attempt by Tory MPs to remove her.

Sir Graham Brady, the 1922 chair, will meet May on Tuesday before another meeting of backbenchers, who are incensed by the prospect of a Brexit deal involving a customs union that could be struck with Labour. He will tell the prime minister she has lost the support of her MPs and that they may be willing to change party rules that protect her from a leadership challenge until December.

Conservative MPs believe the number who have submitted letters of no confidence to Brady tops 100 and could already have reached the crucial number of 157 necessary to force her out of office.

Gavin Williamson, the former chief whip who was sacked as defence secretary on Wednesday, could also be pivotal in attempts to remove the prime minister. Tory Brexiters have rallied to his defence and some are looking to him as a figure to hasten her downfall by making an intervention tearing into her record.

The former Conservative leader Iain Duncan Smith broke cover over the weekend to openly call for May to be forced out of office if she would not name a date to hand over the leadership.

With the prime minister’s position still looking precarious, a parade of Tories lined up at the weekend to display their leadership credentials. Stewart, a centrist who has been defending May’s Brexit deal and her decision to hold talks with Labour, was the only one to be open about his ambitions.

He used an appearance on Sky’s Sophy Ridge on Sunday to say the party would lose 4 million remain voters if it tried to emulate Farage’s Brexit party, which is outpolling the Conservatives for the European elections.

“We’d lose young people, we’d lose Scotland, we’d lose London and we’d lose a lot of the most energetic parts of this country,” he said. “We’ve got to be a broad party. We’ve got to be able to stretch all the way from Ken Clarke right the way through to Jacob Rees-Mogg.”

Other potential candidates to give interviews included Dominic Raab, who set out his backstory in the Sunday Times alongside his wife, Erika. The rightwinger, who would pursue a hard Brexit, appeared to be trying to soften his image, unveiling a policy of preventing women being sacked on maternity leave and allowing fathers two weeks of paid paternity leave. He has been criticised for previously having described feminists as “some of the most obnoxious bigots”.

He also claimed he would like to cut taxes for low- and middle-income families. “I think the basic rate, taking a penny off that, would be talking to the people who need to know we are on their side,” he said.

Sajid Javid, the home secretary, used a speech to the Welsh Conservatives to criticise his party for “viewing the country through a lens of spreadsheets and efficiency savings” while in government.

“Bureaucratic victories mean very little if you live in a community that feels forgotten. And when we succumb to that temptation, we miss the bigger picture,” he said. “That’s why we mustn’t lose sight of why public services like education matter, or of the ways in which good government can profoundly change the course of a person’s life, the way it altered my life.”