The Tory manifesto says elderly people requiring care in their own homes would have to meet the cost - but would be allowed to keep £100,000.
Mrs May says the changes will secure the social care system and ensure fairness "across the generations".
But a Tory think tank says it could be the "biggest stealth tax in history".
John Stanley, of the Bow Group, told BBC News: "The impact on the core vote will be awful - what I call the 'Tory Shire'.
"Those shire Tories who work hard, play the game, live life by the rules. They're going to wake up Monday around the family copy of the Daily Mail asking themselves what on earth has just happened."
Under the Conservative plans, set out in its general election manifesto, nobody who has assets of less than £100,000 will have to pay for care, which can include everything from help with daily tasks, such as washing and dressing, through to round-the-clock support.
Currently anyone with assets of over £23,250 is expected to pay the full cost of their care. If you are in a care home or nursing home, the value of your house can be taken into account.
But that is not the case if you receive care in your own home. Under the Tory plans the value of your home may be factored in, although the money will not be taken from your estate until after death.
This means some people could face losing their home after they die and will not be able to pass it down to their children.
Mrs May has said the policy is fair because it protects elderly people from the fear that they will lose everything to pay for care.
But some Conservative activists say the policy has not been properly thought through and is causing anxiety among elderly voters.
A Conservative candidate, who did not want to be named, told the BBC they had asked party bosses to explain how homes will be valued, when savings are taken into account, and what happens to people who live in shared accommodation.
There are also concerns that it will have an unfair impact on people living in the North and Midlands, where property prices are lower - and that some people may find they can't go afford to return home after hospital treatment.
Conservative candidate Sarah Wollaston told BBC News:  "If you are somebody who has quite a large asset in your home but you might be living on a very fixed low income - that might make it very difficult for you to go home if you couldn't afford special care.
"I think it's very important that there is a period of grace to help them safely get home from hospital."
Labour leader Jeremy Corbyn has accused the Conservatives of "forcing those who need social care to pay for it with their homes," labelling the policy a "dementia tax".
Liberal Democrat health spokesman Norman Lamb said "elderly people the length of Britain will shudder at these care cost proposals".
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Tory plans to shake-up the funding of social care in England are likely to go down badly with the party's core voters, Theresa May has been warned.