Boris Johnson was on Sunday desperately seeking more Conservative MPs to support his comeback bid, as he tried to secure the 100 nominations needed to enter the ballot to become Britain’s next prime minister.

In a blow to Johnson, Suella Braverman, the Brexiter former home secretary, ruled herself out of running and gave her backing to Rishi Sunak, the former chancellor.

“We, as a party, need to change,” Braverman wrote in the Daily Telegraph. “We need to provide leadership, stability and confidence to the British people. We cannot indulge in parochial or nativist fantasies.

She added: “I need a leader who will put our house in order and apply a steady, careful hand on the tiller. That person, for me, is Rishi Sunak.”

Talks on Saturday night between Johnson and Sunak, the former chancellor and frontrunner in the contest, did not result in any “deal” between the two contenders.

On Sunday Sunak confirmed formally that he would enter the contest to succeed Liz Truss as Tory leader and prime minister.

“I have the track record of delivery, a clear plan to fix the biggest problems we face and I will deliver on the promise of the 2019 manifesto,” Sunak said on Twitter.

In an apparent swipe at Johnson’s chaotic style of government, he added: “There will be integrity, professionalism and accountability at every level of the government I lead.”

Sunak already has 133 publicly declared nominations, according to Financial Times analysis, including key rightwingers Kemi Badenoch, trade secretary, and former Brexit minister Steve Baker.

While many Tory MPs would like Johnson to back Sunak to ensure a speedy “coronation” of the former chancellor, uniting the party, the ex-premier still believes he can make a return to 10 Downing Street.

Johnson’s allies claim that he would win in a head-to-head fight with Sunak if the contest goes out to the Tory party membership, and the former prime minister wants his rival to capitulate. “It’s time for Rishi to put on his ‘big boy trousers’,” said one Johnson supporter.

Given that only just over 50 Tory MPs have declared their support for Johnson — in spite of claims from his allies that more than 100 are willing to back him — Sunak’s team do not believe he has the numbers.

Dominic Raab, a Sunak backer, told the BBC’s Sunday with Laura Kuenssberg that there was no case for a deal, pointing out that the former chancellor had “double the number of MPs” supporting him compared with Johnson.

Raab said there had not been a deal agreed on Saturday between Sunak and Johnson where one candidate agreed to stand aside, but said: “What they did have was a very good conversation about the need for unity.”

But Chris Heaton-Harris, Northern Ireland secretary and a Johnson backer, insisted on Sophy Ridge on Sky: “We definitely have enough numbers.”

Johnson urged his supporters to carry on hunting for MPs to back him on an 8am video call on Sunday. “He was in good spirits,” said one Tory MP on the call.

James Cleverly, foreign secretary and an old ally of Johnson, gave his backing to the former prime minister on Sunday afternoon. “I know Boris has learned lessons from his time in No10 and will ensure the focus is on the needs of the country from day 1,” he tweeted.

Jacob Rees-Mogg, business secretary and another backer of Johnson said: “I’ve been speaking to Boris Johnson and clearly he’s going to stand. There’s a great deal of support for him.”

Johnson, who returned from a Caribbean holiday on Saturday, is hoping his team can turn their boasts about surging support among Tory MPs into reality but time is running out.

Nominations for the Conservative leadership close at 2pm on Monday. To enter the contest candidates must secure the backing of 100 Tory MPs out of a total of 357.

If more than one candidate reaches that threshold, Tory MPs would narrow the contest down to a shortlist of two. They would then hold an “indicative vote” on Monday to signal which candidate commanded parliamentary support, before the final decision is put to an online vote of party members.

Under that scenario a new Conservative leader — and Britain’s next prime minister — would be named on Friday. However, many Tory MPs hope the contest does not go out to party members, who picked Truss as prime minister over the summer.

Penny Mordaunt, leader of the House of Commons, has also formally declared her candidacy, but is trailing far behind Sunak and Johnson.

Mordaunt said on Sunday that despite having the support of fewer than 30 MPs she could still win, telling Kuennsberg: “I’m in it to win it.”

Mordaunt insisted she would also secure 100 nominations, but said: “I’m not sharing my canvassing data.” She added: “I’m very confident about our numbers.” She denied exploring a non-aggression pact with Johnson.

If Johnson made the ballot paper and a final shortlist of two, many Tory MPs believe he could win the contest, given he remains popular with grassroots activists.

But Baker, former head of the pro-Brexit European Research Group, said that a Johnson comeback should be avoided at all costs. “It’s a guaranteed nailed-on failure — we cannot allow it to happen,” he said.

Baker and other Sunak supporters point out that Johnson still faces a parliamentary inquiry into whether he lied to MPs about the “partygate” affair. If found guilty, Johnson could be suspended from parliament and could face a by-election in his Uxbridge constituency.

Raab said the partygate affair and questions about Johnson’s character would soon be dominating the news again: “In a matter of days he’s going to see televised witness testimony including his own,” he said.