The health secretary said his rival candidates who had pledged to leave – deal or no deal – by 31 October were making false promises because parliament would block any no-deal departure.

He said the outcome would be a general election, which would be a “catastrophe for my party and extremely damaging for the country”.

Hancock claimed his plan for renegotiation was detailed, “unlike some of the other candidates”, and would involve first putting a deal to parliament that included a backstop with a time limit, to prove to the EU27 that it could pass. He would then restart the negotiations.

The backstop is a device intended to ensure there will not be a hard border between Northern Ireland and the Republic of Ireland, even if no formal deal can be reached on trade and security arrangements.

Hancock told BBC Radio 4’s Today programme his plan had the support of David Lidington, the de facto deputy prime minister, and that Brussels was open to the change.

“It is eminently deliverable by 31 October. Once you have a majority in the House of Commons things can move quickly,” he said. “This isn’t about opening the whole withdrawal agreement – this is about a time limit on the backstop. The EU have the opportunity at the October European council to agree to that addendum.”

Hancock repeatedly argued that Brussels would concede despite its previous position. “They nearly proposed a time limit on the backstop before but they didn’t think the prime minister would be able to get it through the House of Commons. That’s why it’s important to show you can,” he said.

Three more leadership candidates will launch their campaigns on Tuesday: Rory Stewart, Andrea Leadsom and Mark Harper. The final two, Boris Johnson and Sajid Javid, will launch theirs on Wednesday, before the first round of voting by Tory MPs on Thursday.

Leadsom, the former leader of the house, will present herself as an “optimistic yet realistic Brexiteer” who would pursue a “managed exit” from the EU, involving leaving with no deal but with some small side deals in place to prevent disruption – an option the EU has explicitly ruled out.

She will also promise to declare a climate emergency and a major expansion in housebuilding. “Our party has thrived in the past when it has governed as a champion of the people, providing freedom of choice and opportunity, a strong economy and global leadership,” she will say at her launch.

Harper, the former chief whip, is expected to present himself as a clean skin who has not served in the Theresa May administration.

Although he has said he will not rule out a no-deal Brexit, like Michael Gove and Jeremy Hunt, he has said he will not be bound by the 31 October deadline, after which both Johnson and Dominic Raab have said the UK will leave with or without a deal.

At his launch, Harper will say that functioning government “can only be enforced by someone who hasn’t been party to what has been called the ‘worst example of ill-discipline in cabinet in British political history’.”

All 10 candidates will take part in private hustings the vote, with the first round taking place on Tuesday night.

Overnight, Stewart accused some of his rivals – including Johnson, Javid, Raab, Gove and Hunt – of reckless tax and spending offers totalling £84bn which he said risked undermining the party’s reputation.

“We simply cannot make spending and tax cut promises that we can’t keep,” he said. “We cannot criticise Jeremy Corbyn for reckless spending pledges if we start doing the same ourselves. Cheap electoral bribes could cost us dear.”

Stewart’s campaign said Raab had pledged £38.2bn in tax cuts, including raising the national insurance threshold, scrapping stamp duty on homes under £500,000, and a 5p cut in the basic rate of income tax. Johnson’s widely criticised plan to cut the taxes of higher earners was said to cost £14.1bn.

Johnson’s pledge came under fire from Gove, who said the “one thing I will never do as prime minister is to use our tax and benefits system to give the already wealthy another tax cut”. He also used his leadership launch to tease Johnson about his withdrawal from the contest in 2016.

On Tuesday, the former defence secretary Michael Fallon, who has endorsed Johnson, said he hoped the contest would not focus on personalities.

“We’ve all got to work together afterwards. There’s a wide range of candidates but I hope we can agree to leave some of the personal stuff out of this … in the end we have all got to rally round, support the new prime minister, bring the party together again.”

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