Research using focus groups shows that regaining control over UK borders is a top priority with voters and would be much more likely to vote to leave the EU if the powers are not returned to Westminster.
But the voters think the Prime Minister only made his referendum pledge to neutralise the threat from Ukip at the upcoming general election and will fail to see it through because "he won't risk giving us a say".
Education Secretary Michael Gove's former special adviser Dominic Cummings, who has been a notable critic of Mr Cameron since leaving Whitehall.
It focused on groups of swing voters from C2 or D social backgrounds three English marginals, Warwickshire North, Thurrock in Essex, and Hendon, north London.
Overall, the voters said their wages were failing to keep pace with increasing prices and accepted Labour leader Ed Miliband's argument that government is failing to deal with the cost of living, the report found.
Voters believe Mr Cameron has "bad priorities" and is "good for the rich, not us" but Mr Miliband is seen as "wishy washy" and not prime ministerial, it said.
The Business for Britain pressure group, which campaigns for a renegotiated relationship with Brussels, said it was vital that the Government listened to the pubic's concerns over Europe.
Chief executive Matthew Elliott said: "With this debate set to dominate the political agenda for the next few years, it's important that we fully understand the priorities of the voters on this very important issue for the country."