Andy Burnham wants to offer access to free social care services to everyone in England, and will put his plans before voters at the next general election.
The prime minister told the BBC that it would be a system in which "everyone contributes, everyone is covered". He did not rule out tax rises, acknowledging it may be "difficult or unpopular" with some.
Burnham said he wants to set out his "vision" on Tuesday and Whitehall troubleshooter Baroness Casey, who is conducting a review, will advise on the "best way of doing it".