A YouGov poll in Saturday's The Daily Telegraph indicated that 45% of voters would currently say no to the question set by Prime Minister Tony Blair's Labour government.
Blair's government, which backs the constitution, said the exact wording of the question to be submitted would be: "Should the United Kingdom approve the treaty establishing a constitution for the European Union?"
Another survey published on Saturday put the ''no'' vote as high as 56%.
The newspaper's online survey of 1943 adults revealed that 24% would say yes, while 25% remained undecided on the EU document. A further 7% said they would not vote.
Blow to Blair
The Sun newspaper cited a separate poll by website MoneySavingExpert.com which revealed 56% would reject the constitution with 24% in favour.
The daily said the findings would be a blow to Blair and that ministers fear time was running out to turn the tide towards the constitution.
Blair's Labour government is in
favour of the EU constitution
Michael Ancram, foreign affairs spokesman for the main opposition Conservative party, which is against the treaty, told The Daily Telegraph: "This underlines the very real and deserved mistrust of the British people for this attempt to punch them into a European super state."
But Simon Hughes, president of the opposition Liberal Democrats, which favours the constitution, was adamant victory was still possible.
"The more public debate there is, the more the pro-Europeans will benefit," he said.
"There is obviously a lot of work to be done, but victory for the Yes campaign is achievable."
The treaty was signed by all 25 current EU members in Rome on 29 October last year and is set to come into force in November 2006, provided it is ratified by all member states, several of which will hold a referendum.
According to a Eurobarometer poll on Friday, only 11% of EU citizens said they were familiar with the constitution, while 56% said they knew little and 33% declared they had never heard of it.
"There is obviously a lot of work to be done, but victory for the Yes campaign is achievable"
Liberal Democrats president
Eurobarometer's poll found only 20% of Britons in favour and 30% against, making Britain the only EU state where more people give the constitution a thumbs-down than those who support it.