The Tories are expected to trail in the UK-wide poll behind the eurosceptic Ukip.
Despite this Liberal Democrats are concerned that the Conservatives will "overreact" to such a result.
"It's madness", said one senior LibDem. "The results will be a disaster for the Conservatives but they should just admit that now and move on.
"The fear is even though the results will have been obvious for months, they will have a collective nervous breakdown and take an even more hardline position on Europe.
"There will be this massive overreaction."
Coalition tensions over Europe are expected to flare up today as the Tory-sponsored EU referendum Bill faces its first debate in the House of Lords.
The Bill would enshrine in law David Cameron's commitment to an in/out referendum on European Union membership.
But already there have been warnings that it may not make it through the Upper House.
Although government-backed, the Bill is being spearheaded by Tory backbencher James Wharton and there is only limited time for peers to debate private members' legislation.
Some Labour peers have warned that it is the wrong vehicle for changes to the law, that could have such serious constitutional consequences.