Boris Johnson’s plan to overhaul social care could be unveiled as early as next week, as government sources say an impasse over a cap on lifetime care costs may be broken within days.

The Prime Minister, Health Secretary and Chancellor are understood to be pushing to agree the terms of a landmark package by next Thursday, when Parliament rises for summer recess.

The announcement is expected to include a new tax to fund social care and a cap on the amount people pay towards their own care.

Pressure is mounting on Mr Johnson to spell out how he will grasp the growing care crisis ahead of next Saturday, which marks the two-year anniversary of his arrival in Downing Street.

He memorably stood outside Number 10 and claimed in his first address to the nation that he had a plan "to fix the social care crisis once and for all", which critics have been demanding to see ever since.

Reforms proposed a decade ago by Sir Andrew Dilnot, which would cap lifetime care costs and raise the threshold for when the state steps in to help, are understood to remain Mr Johnson’s preference.

It could cost as much as £10 billion, however, prompting wrangling between the Treasury and Number 10 over how much of the cost should be met with tax rises.

In the Queen’s Speech, unveiled in May, the Government pledged to bring forward proposals before the end of the year.

On Thursday night, one government source said efforts were being prepared to get proposals over the line before MPs break for  their summer holidays: "There are discussions going on. There is a will to try and get something announced next week, but there is still a bit of work to do to get to that position."

How social care reform has been pushed back again… and again

A cap on lifetime care costs "is where the discussion is", the source said, adding: "In terms of a Treasury perspective, how much the taxpayer is going to fund, how much people will be expected to fund themselves?"

Sir Andrew, a social care expert, initially recommended capping lifetime care costs for individuals at between £25,000 and £50,000, with the state covering the rest.

Published in 2011, the Dilnot report also recommended increasing the "floor" – or threshold – below which people would receive means-tested state support towards the cost of their care from the current level of £23,250 to £100,000.

At what level to pitch the floor and ceiling in the plan remain "where the detailed discussions are" at present, the government source said, adding that if an agreement was reached, the plan would likely be announced next Thursday.

A second government source said that a breakthrough on social care before the summer recess had been made more likely by Sajid Javid’s appointment as Health Secretary.

They added that he shared the Prime Minister’s determination to finally deliver a solution to a problem that has dogged successive administrations, with next Thursday being suggested among officials as a possible date.

"He’s determined to sort this," they continued.

The Telegraph revealed earlier this month that allies of Mr Javid said he would look at a tax increase to fund an overhaul of social care, as he joined the Treasury in pushing to find a "sustainable" funding solution.

One former cabinet minister said on Thursday night: "The big question is: how much is the Treasury prepared to stump up? They’ve always started with Dilnot, because that’s what is passed into law. It seems a variant on that is the frontrunner now."

The source added: "What worries the Prime Minister is the political response to Dilnot in the past. I am not sure what would be different this time."