Forty-five percent of more than 6,000 people surveyed in Belgium, France, Germany, Hungary, Italy, Poland, Spain and Sweden said they wanted their own vote, and a third would opt to leave the EU if given the chance, poll firm Ipsos-MORI said.
The size of the potential "Out" vote ranges from as high as 48 and 41 percent in Italy and France respectively to as low as 22 and 26 percent in Poland and Spain, the firm said.
"The Italians in particular hope to have their own opportunity to go to the polls on their EU membership, which lends a sense that even if the (British) vote does ... stick with the status quo in June, it will not be the end of the EU's woes," said Bobby Duffy, head of social research at Ipsos-MORI.
Italy's anti-establishment 5-Star Movement has grown into the country's second-biggest political force, and wants an exit from the euro currency zone. France's hard-right National Front party also wants to drop the single currency.
The Ipsos-MORI online poll found that 49 percent of people in the eight countries thought Britain would vote to leave the EU on June 23, higher than the number in Britain itself, which stood at 35 percent, the survey showed.
And 51 percent said a so-called Brexit would hurt the EU's economy, while only 36 percent thought it would hurt Britain's.
British "Out" campaigners have said the country would be in a strong position in any negotiations on a new trade deal after leaving the bloc, given its standing as the world's fifth-biggest economy.
The "Out" campaign has also warned that the EU is destined for deeper political union, but the Ipsos-MORI poll showed few voters in Europe think that is likely.
Just over 20 percent of respondents in all nine EU countries covered by the survey, including Britain, thought there would be more integration by 2020 compared with 40 percent who thought there would be less.
Forty-eight percent of voters thought a Brexit vote next month would result in other countries also leaving the bloc, compared with 18 percent who disagreed.
The poll was conducted between March 25 and April 8.