Senior Tory MP David Davis is calling for a double referendum - one to approve a list of powers for the UK to seek to seize back and then an in/out public poll once they had been negotiated.
Mr Davis, a former Europe minister and leadership rival to Mr Cameron, said the first vote should be held within a year, before the next elections to the European Parliament in 2014.
Having a renegotiation list backed by a public vote - he predicted it would get the support of 70% of electors on a massive turnout of 90% - would be a "huge negotiating lever" in Brussels, he said.
"We have got to, somehow, dramatically change our relationship with Europe. Not a little bit of a power here and a little bit of a power there. We have got to bring back lots of powers," he said.
More top news
Drug gangs in Mexico City commonly dump the corpses of their victims in public as a way of sending warnings.
Theresa May still wants to press on with her plan to trigger Article 50 by March, which would mean Britain could leave the EU by March 2019.
Donald Trump's spokesman says the president's claim of mass voter fraud was based on "studies and evidence", but refused to give details.