May said the extra funding will help "secure" the NHS's future.
But the PM is under continuing pressure to explain where the money will come from to pay for the rise.
In meetings with Philip Hammond, he stressed the chronic needs of the NHS and the importance of every tenth of a percentage point, before withdrawing to let Mr Hunt and the chancellor thrash out the detail.
She did not reveal how all the money would be raised, but said: 'Some of the extra funding I am promising today will come from using the money we will no longer spend on our annual membership subscription to the European Union after we have left'.
In an interview with Tom Swarbrick, May's ex-head of broadcasting in Downing street on LBC radio station, the prime minister announced that the the United Kingdom will be able to spend "around £600 million a week" more on the NHS by 2024, in what ministers are calling a "70th birthday present".
Analysis published by the Institute for Fiscal Studies (IFS) last month said that total health care spending in England (not just on the NHS) would need to increase by around 3.3% a year over the next 15 years in order for the NHS to keep functioning at its current level.
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'So we will do more than simply give the NHS a one-off injection of cash.
She said she wanted to see better mental health services and cancer survival rates.
The Conservative chair of the House of Commons' Health and Social Care Committee, Sarah Wollaston, said the idea of a Brexit dividend was "tosh".
The prime minister said this will be partly funded by a "Brexit dividend" - money Britain will supposedly save from no longer contributing to the European Union's budget.
"We are absolutely confident that we will deliver a Brexit deal that will.be good for the United Kingdom, good for our European friends and partners". There has been some talk of a "Brexit dividend", although extra taxes and borrowing will be needed as well. To deliver, this plan must be about more than money.
The Scottish politician noted the NHS is a pillar of British society and praised Theresa May for making it her top spending priority.
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She said that meant that "at the end of five years, by 2023-24, there will be £20bn more in real terms being spent on the NHS".
She attacked the Cameron government's key health reform - driven through by the former Health Secretary, Lord Lansley - which created clinical commissioning groups to negotiate legal contracts with other parts of the NHS.
"With this extra spending power, Northern Ireland now has the opportunity to improve health outcomes for our people and create the NHS that they deserve".
"We have looked carefully at what we have put into the NHS to ensure that we deliver world-class healthcare".
In an interview with The Sunday Times, health minister Jeremy Hunt said: "There's a total understanding across the government that the health and social care systems are interdependent and that to make this settlement work we also need a long-term plan for the social care system as the next step".
"But we've seen less than half of that in Scotland; the SNP have not passed on health consequentials".
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