The Health Secretary told BBC Radio 4's Today programme that they 'are clear that there will be an increased burden of taxation'.
She added: "The Tories" claim that this is funded by a "Brexit dividend' is simply not credible as the United Kingdom will be paying a £40bn bill to leave the EU".
May has pledged to increase funding for the National Health Service by 20 billion pounds ($26.57 billion) by 2023/24, even as some critics said that Britain's departure from the bloc will weaken, not strengthen, public finances.
Speaking to the BBC's Andrew Marr Show, she said the government is working to secure the future of the NHS.
"It must be a plan that tackles wastes, reduces bureaucracy, and eliminates unacceptable variation, with all these efficiency savings reinvested back into patient care".
Appearing on the BBC's Andrew Marr show on Sunday, Mrs May said the money Britain now pays into the European Union could be used for the NHS in the future.
Plus, the funding increase announced focuses only on NHS England.
May's announcement may be more about saving the Conservative party, but Chancellor of the Exchequer Philip Hammond will still have to come up with the cash.
The prime minister is facing a backlash for linking her NHS cash boost with the Vote Leave pledge during the referendum campaign to spend Britain's European Union contributions on the health service instead.
Theresa May has reiterated her much-questioned belief that a new funding deal for the NHS can be financed in part using money saved thanks to Brexit, but warned that taxpayers would also have to expect to contribute more.
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Conversations about residents returning home have taken place between the incident command post and law-enforcement agencies. Due to the 416 and Burro fires, manpower is already thin. "We recognize the conditions are dry and hot", Gov.
She continued: "despite more funding, more doctors and more nurses, and great progress on treatments, our NHS is under strain".
"The devil is in the detail and the Tories have form of cutting other budgets when they are passing on consequentials which leaves the Scottish Government less than the health consequentials from the UK Government", she said.
The DUP's Simon Hamilton said the announcement was a "hugely welcome boost for our National Health Service".
The PM will also recall her own reliance on the NHS for help when she was diagnosed with type one diabetes, saying: "I would not be doing the job I am doing today without that support". But it's right that we must recognise the impact social care has on the NHS, and the way in which these work together.
He said: "This programme of investment and reform was supported by a clear and specific increase in national insurance - unlike the plans announced today which appear to be dependent on mystery tax increases and a mythical Brexit dividend the IFS confirms can not fund the extra spending".
The NHS marks its 70th anniversary this year.
That's a faster increase than in recent years, .
"I'll be making sure we keep the pressure on the SNP to deliver for Scotland and deliver for our NHS".
Theresa May said some of the spending boost would be the result of a "Brexit dividend" as the United Kingdom would soon no longer be sending large sums to Brussels.