The former Conservative cabinet minister and Brexit rebel Justine Greening has announced she will quit as an MP at the next election and criticised her party for becoming “the Brexit party”.

Issuing a stinging criticism of Boris Johnson’s government and its lurch further to the right, she said the Conservative party was “narrowing down its appeal” to people across the country.

The former education secretary is among the group of Tories, including the former chancellor Philip Hammond, who face being thrown out of the party because of their intention to block no deal in parliament this week and delay article 50.

“I have sent a letter to the prime minister saying I will not be standing as a Conservative candidate at the next election,” Greening told BBC Radio 4’s Today programme.

“My concerns about the Conservative party becoming the Brexit party have come to pass.”

Greening, who was brought up in Rotherham, was the first comprehensive-educated Conservative education secretary, and the first gay female cabinet minister. She also served as international development secretary and for a time was considered one of the most senior women in government.

She is part of the so-called “Gaukeward squad” of Conservative MPs who want to stop no deal and among the handful of Tories who publicly back the People’s Vote campaign for a second referendum.

Her constituency of Putney voted heavily for remain in 2016.

On the prime minister’s threat of holding a snap general election, she said: “I don’t believe the Conservative party will offer people a sensible choice at the next election in respect of the fact Boris Johnson is going to offer people a general election that offers them no deal or Jeremy Corbyn.”

Johnson threatened to call a general election on 14 October if MPs voted for a rebel motion to legislate against Britain crashing out of the EU without a deal.

Greening said: “That is a lose-lose general election for Britain and I think a far better way of resolving the path forward on Brexit is to give the British people a direct choice between the different options on Brexit themselves rather than a messy general election, which I believe all the evidence suggests will be yet again inconclusive on a route forward on Brexit.”

The foreign secretary, Dominic Raab, said on Tuesday he wanted to “focus minds” of MPs considering voting against the government to block a no-deal Brexit and to highlight the consequences of backing a delay.

He told the Today programme: “We want to focus minds of all MPs that a deal is within sight that is faithful to our manifesto commitments but also to the referendum and any more delay … what does anyone realistically think a delay would do, rather than strengthen the hand of Brussels and weaken the hand of our negotiators?”

He said repeatedly the Conservatives did not want a general election.

On Tuesday, Corbyn will meet a group of opposition party leaders in his office, including Ian Blackford from the SNP and Jo Swinson from the Lib Dems, for further talks on how they might block no deal.

There has been significant confusion in the past 24 hours about whether Labour would vote in favour of a general election, even if triggered by the prime minister, or whether it is prioritising no-deal legislation.

The shadow cabinet minister Tony Lloyd told the BBC’s Newsnight the party would not back a general election triggered by Johnson.

The shadow attorney general, Shami Chakrabarti, told the Today programme: “It’s about sequencing. We need to ensure we get this legislation locked down … and guarantee Britain would not crash out of the EU in an election campaign period. We’ve also got to try as best as possible to ensure that it wouldn’t be possible for the sitting, squatting, prime minister in this period to set a general election, then change the date.

“Then if we get the sequencing right, and get sufficient support across the house, then we live and breathe a general election.”

• This article was amended on 4 September 2019. An earlier version incorrectly referred to Greening as the first comprehensive educated education secretary. To clarify: she is the first comprehensive educated Conservative education secretary.