Ministers are considering a national insurance rise – described as a social care and health levy – in order to overhaul the UK’s social care system, as Boris Johnson refused to recommit to the Tory manifesto promise not to raise taxes.

The prime minister declined to reassert the pledge not to raise income tax or national insurance when asked about plans for tax hikes at the Downing Street press conference on Monday. Johnson hinted a proposed offer was imminent, saying: “It won’t be too long, I assure you,” though Whitehall sources have suggested no deal has been reached.

No 10 and the Treasury have also examined plans for an income tax rise on over-40s, but one government source said plans had moved away from that in recent days because of concerns the rise would need to be substantial in order to meet the costs.

Rishi Sunak is understood to have been clear that any potential tax rise must be agreed and announced at the same time as the new policy on care costs, likely to include a lifetime cap at a level yet to be agreed. One Whitehall source said that made it less likely the agreement could get over the line before the summer recess.

Former health secretary Jeremy Hunt backed an income tax rise on Monday, which he called a health and care premium, saying a 1% rise would generate £6bn in income that would help tackle catastrophic hospital backlogs and other healthcare needs.

Whitehall sources suggested ministers were leaning towards a national insurance rise, but either would break the spirit of the Conservative “triple tax lock” manifesto promise, though the rebrand might help sell the policy to MPs.

Increasing national insurance by 1p for employees and the self-employed would raise around £6bn a year, according to calculations by the Resolution Foundation.

Hunt, who chairs the health select committee, said there was “a growing realisation that with the Covid backlog we’ll never get the NHS back on its feet without social care reform”.

He said there were other options to raise money for the exchequer but none were particularly palatable. The concept of a social care precept already exists for council tax, with local authorities permitted by the Treasury to use it to raise money. Hunt said removing the cap was a possibility, but said “soaring council tax bills to pay for social care” would affect public support.

He said removing the exemption for pensioners to pay national insurance would be unpopular and only raise half a billion pounds a year.

“The attraction of a health and social care levy is it would fund the NHS backlog in the short term and desperately needed improvements in the social care system in the medium-longer term,” Hunt tweeted. “It would also be transparent about the need for resources and capacity in both sectors.

“A health and care premium is the most honest solution, with a sensible debate on whether we’ll fund the new dementia drugs on the way, make our cancer survival rates as good as Denmark/Australia – and proper social care – and how much that’s about funding [versus] innovation/efficiency.”

One source close to the discussions said putting the increase on national insurance would put the government at risk of accusations that it was putting a penalty on younger working people and particularly low earners. “It’s definitely on the table, but it does target those hit hardest by the pandemic,” the source said.

The Treasury has traditionally opposed hypothecated taxes, where tax rises are announced with the revenue determined for a specific department, but Sunak is said to be reconciled to branding any rise destined to tackle social care.

Treasury and No 10 sources said there was still no timetable for the announcement, which has been widely expected this week, promising only that a plan would be set out by the end of the year.

Torsten Bell, chief executive of the Resolution Foundation, said: “Tax rises will be needed to deliver decent social care, but a national insurance rise is a terrible way to raise the funds required. It’s a tax disproportionately loaded on to younger and lower-paid workers, compared to a fairer rise in income tax.

“Why we would target a tax rise on the groups who have been hardest hit by the economic impact of this pandemic, while exempting older and wealthy individuals, is completely beyond me.”