Boris Johnson’s chances of securing parliamentary approval for his Brexit deal were boosted after ex-cabinet minister Amber Rudd said she would vote for it and expected most of the Tories expelled from the party over a previous Brexit rebellion to do the same.

The former home secretary, who walked out of government and quit the Conservative whip in protest at the mass expulsion of 21 colleagues, said she believed Mr Johnson’s deal would now get through parliament in the coming days, though she acknowledged the coalition behind it was “very fragile”.

Her announcement marks a significant gain for the prime minister, who will need the votes of the bulk of the ex-Tory independent group to get a majority for the withdrawal agreement sealed with EU leaders last Thursday.

Despite backing Oliver Letwin’s amendment withholding parliamentary approval on Saturday, Ms Rudd told Sky News’s Sophy Ridge on Sunday that she would have backed the deal itself if it had been pushed to a vote.

She said Mr Johnson’s package was not as good as the deal struck by his predecessor Theresa May, which she also supported.

“I support the prime minister’s deal and I have told him ‘I will support you next week’,” said Ms Rudd.

“I want to support it and I will and I think, not all of us but most of the former Conservatives who supported the Letwin amendment will do so as well.”

There were also signs that Mr Johnson could pick up votes from among a group of 19 Labour MPs, mostly from Leave-voting seats, who wrote to the EU earlier this month calling for a deal.

One of the group, Lucy Powell, declined to say which way she would vote but told Ridge: “It’s clear from what Amber Rudd was saying, and others, that the votes probably are there for a deal.”

Ms Powell said that many Labour-supporting communities had voted leave and said she “worries for the future of our party” if it turns its back on them.

“That’s why we’ve go to work really hard now to try and respect the outcome of the referendum but shape that Brexit deal, get the best possible outcome that we can over the coming days,” she said.

Foreign secretary Dominic Raab said he was now “confident” that Mr Johnson’s deal will receive parliamentary approval.

Mr Raab told BBC1’s Andrew Marr Show: “We now – notwithstanding the shenanigans in parliament – appear to have the number to get it through.

“We believe we’ve got the numbers and we’ll keep talking to the DUP and see if there’s further reassurances that can be provided to them.”

The prime minister’s no-deal supremo Michael Gove said that he was so confident of securing the deal that he had placed a bet with the health secretary, Matt Hancock, on the result – though his office later clarified that the wager was on the size of Mr Johnson’s eventual majority and not on whether he would win.

Mr Gove told Ridge: “We are going to deliver by 31 October. We are going to ensure that we get this deal done.

“I’m confident that with the support of good people, with whom we may have disagreed in the past but who respect democracy, we will get this deal done.”

Mr Johnson’s chances of success are dependent on retaining the support of around 30 Brexit hardliners in the European Research Group, who voted down Theresa May’s deal but backed him over the Letwin amendment.

Nigel Farage slams Boris Johnson's 'rotten' Brexit deal as he calls for extension in order to have election

Brexit Party leader Nigel Farage urged ERG members not to vote through the “rotten” deal, calling instead for short extension to allow for an early election.

Mr Farage told Sky: “I warn everybody, if this treaty goes through, nothing will have changed at all and I think far better to have a short delay and a general election where we might solve this because we’ve got a Leave country – that hasn’t changed one little bit – and a Remain parliament. A general election could just sort this out…”

Dismissing Mr Johnson’s package, he said: “This is not Brexit ... This is a rotten deal. If Boris Johnson had presented this six months ago to the House of Commons, the ERG – every man and woman – would have voted against it.

“I do understand, because of Brexit fatigue and anger in the country, the temptation to vote for it but it is nothing more than Brexit in name only, it will not solve anything, this will not end things.”

Mr Farage said he was still open to an electoral pact with Johnson if the prime minister ditched his withdrawal agreement.