The NHS will get more money over the next three years than the Treasury had planned after last-ditch lobbying for a bigger budget by the health and social care secretary, Sajid Javid.

Whitehall sources said the chancellor, Rishi Sunak, had been persuaded to increase the funding he intended to hand the NHS by at least £1bn, taking the expected rise from £5bn to at least £6bn a year.

In a separate move, the government confirmed on Monday the NHS in England will be given an extra £5.4bn over the next six months including money specifically targeted to help tackle the backlogs caused by the pandemic.

Concern over NHS funding was heightened last week when two powerful bodies made clear the health service in England needed no less than an extra £10bn a year to cope with Covid-19 and the huge backlog of care.

The Guardian reported last week that Sunak was not prepared to give the NHS more than another £5bn a year – half of what was being demanded by NHS Providers and the NHS Confederation, which represent England’s 213 health care trusts.

Whitehall sources with knowledge of the discussions around the future funding of both the NHS and social care said the health secretary had had some success in convincing his successor at the Treasury to move beyond a supposedly final £5bn rise.

Boris Johnson is due to make a major double announcement on Tuesday setting out how much money the NHS will get in 2022-23 to 2024-25 and the detail of his hugely controversial plan to “fix” social care.

Ministers hope the extra money that Sunak has approved will encourage the two NHS organisations to endorse the government or at least not criticise it too loudly.

However, those bodies made clear in a joint report last week that patients would be put in “peril”, services would have to be cut, the backlog would balloon further and quality of care would decline unless the service received the full £10bn.

On Monday, the government confirmed the NHS in England would get £5.4bn extra for the second half of the financial year to help to clear the waiting lists faced by patients due to Covid-19.

Of the £5.4bn, £1bn will be spent tackling the backlog as the total number of people waiting for hospital care, especially surgery, has reached 5.45 million – the highest figure since records began.

Javid has warned that waiting lists could hit 13m as growing numbers of people who did not or could not access NHS care over the last 18 months see a GP.

“The NHS was there for us during the pandemic, but treating Covid patients has created huge backlogs,” said Johnson. “This funding will go straight to the frontline, to provide more patients with the treatments they need but aren’t getting quickly enough.”

Another £478m will be used to speed up the discharge of hospital patients who are fit to go home, in a move that the Department of Health and Social Care (DHSC) said would “free up thousands of extra beds and staff time to help the NHS recover services”. There are fears that a further wave of Covid and a bad winter could overwhelm hospitals, which usually run at much higher bed occupancy than the 85% experts say is safe.

Anita Charlesworth, head of research at the Health Foundation thinktank, welcomed the £5.4bn but cautioned that it was “only the first instalment of the substantial funding needed to put the NHS on the road to recovery.”

The NHS will put a further £500m into increasing the capacity of operating theatres and into “productivity-boosting technology”, so more procedures can be undertaken, the DHSC added.