Boris Johnson’s allies will switch their focus to winning a vote of no confidence, after conceding that they now have little chance of stopping one being triggered.

The prime minister will this week launch a health and housing policy fightback in a last-ditch attempt to win over his critics. He is widely expected to face a vote on his leadership as soon as this week, with some MPs predicting that the threshold of 54 letters asking for one has already been exceeded.

In a remarkable shift in tone, the business minister Paul Scully acknowledged on Sunday night that a vote of no confidence “might well happen”, but insisted Johnson would “face it down”. “Whatever happens, we’ve got to get back to governing, to tackle the things that people want us to do on a day-to-day basis.”

Hours earlier, the transport secretary, Grant Shapps, had said he did not think there would be a vote this week.

A No 10 source said Johnson would demonstrate over the coming days that he was “getting on with the job” and acknowledged it was also aimed at showing his determination not to be fatally weakened by a confidence vote, which the prime minister expects to win narrowly.

The source said there was no prospect of Johnson offering his resignation if he won by only a small margin, or of offering to go before the next election, as Theresa May did.

Johnson will not shy away from the potential humiliation of the two upcoming byelections, aides said, and is planning visits to Wakefield and Tiverton, both of which the Tories are widely expected to lose to Labour and the Liberal Democrats respectively.

Some newer MPs are said to be nervous of acting too soon and are considering pushing to delay a confidence vote until after 23 June, when the byelections are due to be held, to give the best chance of ousting Johnson and allow potential leadership candidates more time to prepare.

Amid accusations from some of Johnson’s supporters of No 10 complacency, the prime minister will focus this week on the NHS backlog, heralding the progress the government is making to tackle waiting lists, in a nod to the use of cash raised by tax increases that are unpopular in sections of the party.

Johnson will announce on Monday that one million checks and tests have been carried out since the rollout of new community diagnostics centres, freeing up hospital capacity.

He explicitly linked it to the national insurance rise dubbed the health and social care levy, much criticised by Tory MPs, and said the funding would mean the NHS could “tackle the Covid backlogs, reduce waiting lists and save lives”.

Later in “health week”, a major review of NHS management by the former vice-chief of the defence staff Gen Sir Gordon Messenger will propose an overhaul of NHS leadership structures to helping failing trusts emulate those that are performing best.

Sajid Javid, the health secretary, has pledged that the review will be the most far-reaching of leadership in the NHS for 40 years and will “level up” regional disparities in care.

This week Johnson is also expected to announce an extension of the right to buy to millions of people who rent from housing associations, as well as the extension of other home ownership schemes. There are also tentative plans to formally table the controversial legislation to override parts of the Northern Ireland protocol.

Polling over the weekend showed the Conservatives on course for a humiliating defeat in Wakefield. Several MPs from the 2019 general election intake have told colleagues they will not submit a letter until after that result. “The red wall MPs who are wavering are looking only at Wakefield,” one MP said. “Only then will the penny drop that he is not actually popular at all.”

When a confidence vote comes, Johnson’s allies say it will be crucial for him to show he has the support of the majority of backbenchers.

“Theresa May lost the backbenchers,” one MP said. “The prime minister has always got to get the payroll votes [those holding government posts] on side. But what he needs to do is get half the back benches at least, because then that gives a strong signal.”

One former minister, a supporter of Johnson, said that even among those who back him he will need to win back trust and show he can win the next election with a big agenda.

“He will survive, but obviously that will weaken him. If he soldiers on, then he’s got an awful lot of work to do to make good [the] damage that has been sustained. Knowing Boris, he might well be able to find his way.”

But rebel MPs have circulated a private briefing document setting out a stark electoral picture about Johnson’s prospects. It says the booing of Johnson at the Jubilee “tells us nothing the data does not” and that no social group polled says they trust the prime minister.

Another point says the “entire purpose of the government now appears to be the sustenance of Boris Johnson as prime minister” - pointing to his personal negative ratings and saying that “defending the indefensible” is not to protect the party but one man.

One minister said the residual loyalty to Johnson was now very thin even among his supporters. “Of course, stuff is going to catch up with him. He’s toast. Everyone is tired of the drama. The only question is whether he manages to get through the election and to be able to get a bit longer in No 10 before we get rid of him. We won’t stand this shit for ever.”

Just 18 MPs have publicly declared that they have sent a letter, but MPs who are running the numbers believe there are at least 70 who have now publicly expressed a lack of faith in the prime minister.

Most MPs seem resigned that the dam will break, but the timing is virtually impossible to guess given the lack of coordinated effort. “It’s all about individual MPs. There’s not even any WhatsApp groups as far as I know,” said one MP who opposes Johnson.

Most MPs are prepared to bet that a challenge is imminent. “I’d say we were already there, and when Graham Brady [the chair of the backbench 1922 Committee, who receives the letters] gets back to his office on Monday, there will be a load more,” the MP said. “I’d expect the vote on Wednesday.”

The vote is a secret, in-person ballot held in parliament. To survive in office, a Tory leader requires the backing of a minimum of half of his MPs plus one, meaning Johnson would need the support of at least 180 of his parliamentary party.

“Getting to 180 is a big ask, but it’s a secret ballot,” one MP said. “I think a third of the payroll could go against him. If it’s a third of them, and two-thirds of backbenchers, suddenly you’re in business.

“There will be a coordinated ring-round by No 10, but I think that if we get to 180 or not, the number voting against will be a lot higher than the PM might think.”