Councillor Izzi Seecombe, of the Local Government Association, said: "Without essential council services, which help people live healthy lives in their own homes and communities, the NHS can not thrive".
"As we leave the European Union and stop paying significant annual subscriptions to Brussels, we will have more money to spend on priorities like the NHS", May said in a post on her Facebook account.
The Government said that under the plan by 2023-24, the NHS budget will increase compared to today by over £20 billion a year in real terms, which is approximately £600 million a week in cash terms.
Theresa May told the BBC that a 10-year spending plan, due to be formally released on Monday, will mean "significantly more money going into the NHS".
"As we celebrate the fact that our NHS is 70 years young, it's important we look to how we deliver the sustainable health and social care communities across Scotland will require in the future", she added.
There will also be more cash to cover a pensions gap in the NHS.
The National Health Service is to get an extra £20bn per year.
The government has yet to confirm a plan for finding up to £11bn of the money promised yesterday, with the PM telling Swarbrick that tax rises are on the way in the autumn budget.
Theresa May has announced that the NHS will be getting an extra £394 million a week after Brexit- but your taxes will also be going up.
The prime minister was joined at the announcement by her chancellor, Philip Hammond and health secretary Jeremy Hunt. However, because of United Kingdom spending commitments made during Brexit negotiations, no such "dividend" will exist until at least 2022.
We've written before about the concerns raised by think tanks and NHS senior managers that funding specifically targeted at the NHS has come at the same time as other elements of health spending, for example public health, have been falling.
Appearing on the BBC's Victoria Derbyshire show on Monday morning, he said: "It is freaky that the government is deciding to bring this very toxic, divisive debate back into the NHS proposals".
"Labour would have invested almost £9 billion extra this year in the NHS and social care, while asking the wealthiest and big corporations to pay their fair share of tax".
With so much at stake, it is only right that the public are given the chance to take a view on the final Brexit deal through a people's vote. And I'm very clear that we need to make sure that in dealing with social care it doesn't put added pressure is on to the NHS.
"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.
"But we've seen less than half of that in Scotland; the SNP have not passed on health consequentials".