A new poll by TNS has found that 51 per cent of Scots want to remain as part of the 28-member bloc, with just 19 per cent wanting to leave.
However, the remaining 30 per cent of voters are undecided, as voters are focussing on the upcoming Scottish parliamentary elections, also taking place this summer.
"The high number of undecided voters at this stage is not surprising, given the greater focus in Scotland on the Holyrood election," said Tom Costley, head of TNS Scotland.
He added: "This means the debate on EU membership is not yet taking on the same level of importance as in other parts of the UK. However, the anticipated high turnout may also show the public appreciates the importance of the issue, given it is being put to a referendum."
This contrasts to a recent poll by OBR that found just five per cent of British voters were undecided, indicating the result of the referendum will turn on which camp can get its supporters to turn up and cast a vote on the day.
Still, turnout for the referendum in Scotland is set to be extremely high. Some 72 per cent of voters said they were certain to vote.
The poll also found those planning on voting Labour in the Scottish election are the most likely to vote to remain, as 66 per cent of them back the UK's continued membership of the EU.
Perhaps unsurprisingly, Conservative voters commit the lowest level of support to remain. Some 33 per cent of their voters want to vote to leave, with just 45 per cent backing remain.
While 60 per cent of Scottish National Party voters back remain, 24 per cent of them are also undecided - the largest level of all the parties.