Spending on the NHS should also be paid for by a dedicated tax marked on every payslip, the former health minister suggested.
Under Mr Lamb's plan, taxes would not be increased as the new levy would be offset by deductions to income tax or national insurance.
He has warned the NHS faces collapse without an urgent cash injection.
The plans are not yet party policy and will not be put to this year's conference in Bournemouth.
But Mr Lamb, the party's health spokesman, told party members he was "very interested in the idea of a dedicated NHS and care contribution - separating it out from the rest of taxation, clearly identified on your payslip.
"And I am really interested in the idea of the right for local areas to raise additional funds for the NHS and care if they choose."
The Lib Dems say he would like to implement the ideas across the UK, although, as health and social care are devolved, it is unclear how this would be enforced.
Mr Lamb - who lost out to Tim Farron in a leadership election in July - proposes a cross-party commission to explore the ideas. He intends to consult health bodies and professionals, patients, trade unions and academics.
Ministers have pledged £2bn in this financial year for the NHS, and an extra £8bn by 2020.
But Mr Lamb told the BBC that this was insufficient and, having "seen the books" as a minister in the last government, he feared the NHS could face a funding shortfall of £30bn by 2020.
"The bottom line is with rising demand because of an ageing population we need more investment," he said.
Mr Lamb also warned that the social care system was "on its knees" and could collapse without a cash injection of £5bn.
"I've been in the department. I have seen the books and I am deeply concerned. If we carry on regardless, the system will crash."
Taxpayers are already shown how much they have contributed to the health service in annual personal tax statements.
An attempt to establish a cross-party commission on social care before the 2010 election - led in part by Mr Lamb - collapsed in acrimony.
English councils should be allowed to put up taxes to fund the NHS, Norman Lamb has told the Lib Dem conference.