Boris Johnson will tell mutinous MPs that the healthcare system is in crisis and they “cannot expect it to recover alone” as he faces a growing red wall rebellion over tax rises.

One Conservative frontbencher told the Guardian they were considering their position over a planned national insurance rise to fund an overhaul of social care and tackle the NHS backlog.

They questioned the point of serving a government that was not pursuing the 2019 Tory manifesto, with a second pledge, on the pensions triple lock, also set to be broken. A series of MPs from former Labour red wall seats complained that the tax increase would hit workers in their constituencies while leaving pensioners untouched.

On Tuesday the prime minister will frame the tax increase, which could raise up to GBP10bn a year with a 1.25% rise in national insurance contributions (NICs) for employers and employees, as essential to combat the NHS waiting list crisis.

In the long term, funding will be used for social care costs once a patient reaches a costs cap, thought to be about GBP80,000. Under the current system, anyone with assets of over GBP23,350 funds their care in full, and about one in seven people pay over GBP100,000.

After significant cabinet wrangling, plans were signed off on Monday night for the new social care and NHS funding package which Johnson will present to cabinet on Tuesday followed by a statement to parliament.

Cabinet ministers had had no sight of the plan on Monday night, suggesting the signoff would be perfunctory. Johnson, the chancellor, Rishi Sunak, and the health secretary, Sajid Javid, will present the plan at a joint press conference.

Ahead of the announcement, the prime minister will warn MPs that the NHS is at crisis point. “The NHS is the pride of our United Kingdom, but it has been put under enormous strain by the pandemic. We cannot expect it to recover alone,” Johnson said in comments released overnight.

“We must act now to ensure the health and care system has the long-term funding it needs to continue fighting Covid and start tackling the backlogs, and end the injustice of catastrophic costs for social care. My government will not duck the tough decisions needed to get NHS patients the treatment they need and to fix our broken social care system.”

The government could bring a snap vote on the proposals as early as this week to nip the prospective Tory rebellion in the bud, by proving those opposed to the policy do not have the numbers to inflict defeat, and to prevent more unfavourable press coverage.

After months of negotiations with the chancellor, Johnson is prepared to order a rise in NICs, favoured as it will tax both workers and businesses. But critics say it will disproportionally hit younger workers who will pay to allow rich pensioners to keep more of their assets because NICs are not paid by pensioners. In Scotland, where care costs are largely paid by the state, voters are likely to bear an additional tax burden which the SNP have termed “the new poll tax”.

Labour is also expected to oppose the rise, but has not yet made a final decision about how it would vote if the plan is brought to parliament. Neither has Keir Starmer said how his party would boost resources for the creaking system. Some shadow cabinet ministers are privately frustrated that the party has not produced its own proposals.

However, the shadow social care minister, Liz Kendall, said the focus on the cap was failing to address wider problems with social care. “People want to focus on the cap and how it would be funded and Labour’s position, but all of the users of social care are saying: ‘This is nothing about what we want,'” she said, pointing to low-paid staff and the lack of support for unpaid carers. “The cap alone won’t fix the crisis.”

The fiercest criticism of the plan has come from the prime minister’s own party, including privately from cabinet ministers. In public, Johnson is facing his first major red wall rebellion from Tory MPs.

Jake Berry, leader of the Northern Research Group of Conservative MPs, said he did not think it was reasonable for people in his constituency – more likely to be on lower wages than those in affluent southern England seats – to have to pay more tax to support those who simply want to “keep hold of their houses in other parts of the country where house prices may be much higher”.

“It doesn’t seem fair to me, particularly after this pandemic, where so many people have taken great sacrifices to keep people safe – it’s particularly hit the youngest, particularly hit those in work – that we then ask those in work to pay for people to have protection in care,” he told the BBC.

Dehenna Davison, the MP for Bishop Auckland, said the Conservatives “absolutely cannot go against this manifesto”.

Alex Stafford, who represents Rother Valley in South Yorkshire, said he hoped taxes would not be raised “willy-nilly” without concrete plans for how the money would be used. He said: “My concern is if they just add an extra 1% on national insurance or whatever, but no actual fundamental way to make social care provision better, it’s a bit pointless … We can’t just raise it without a new way of providing social care.”

One former minister also said Johnson was “trying to appease everyone – and leaving us all disappointed”, adding that Downing Street had become “an utter car crash to the point it’s really starting to stretch our patience”.

A second took aim at Johnson, claiming raising NICs was the latest example of him appearing to abandon key Conservative principles. They said: “It’s like being in a clown car with a very strange driver veering relentlessly from the left to the right hand side of the road constantly. I voted for him. I like him. But he won’t drive the UK over a cliff with his principle-free impetuousness. This has got to stop.”

At a private backbench reception on Monday night, Sunak told colleagues it would be “a tough autumn ahead … That doesn’t mean there won’t be disagreements – there always are – but we should never lose sight of the central fact that we are a team.”