The survey of Tory constituency chairmen also indicated a lack of consensus on what the PM's negotiations with EU leaders had achieved.
The programme approached all 330 chairmen in Tory constituencies.
Of the 128 who replied, 54 said they would vote for the UK to leave the EU.
About a quarter, 31, said they would vote for the UK to remain and a third said they had yet to make up their minds.
Asked about Prime Minister David Cameron's renegotiation of the terms of British membership, 61 respondents - almost half - said they were not satisfied with the proposed new deal with the EU.
About a third, 40, were satisfied.
One of the chairmen said Mr Cameron "should have bought himself a handbag," implying the prime minister didn't measure up to Lady Thatcher when it came to negotiating with Brussels.
But "he did his best", several others said.
"I'm still hungry for facts," said one chairman, reflecting the view of several still weighing up which way to vote.
Some hinted at a relative indifference to the issue among voters. "The subject of Europe rarely comes up on the doorstep," another said.
Others highlighted divisions, even within themselves. "My heart says leave; my head says stay," said one.
So what does this tell us?
Let's be honest first about what it doesn't tell us: it doesn't tell us anything definitive, because lots of Conservative constituency chairmen didn't want to talk to us, and we have no idea what their views are.
And we only approached constituency chairmen. That leaves many, many loyal Tory activists whom we didn't even ask to speak to.
But, with those caveats inserted, what we have found out does offer a snapshot of a conflicted party.
Beyond the numbers, it is interesting to take a look at what those constituency chairmen are thinking.
Their reflections, in private, are revealing.
Conservatives can take comfort that their range of views reflects those of the country at large.
But they are painfully aware too that having exposed these differences, they will have to carry on governing after the referendum - with a big chunk of their tribe wounded and defeated, regardless of the outcome.
