A survey of 1,100 people – by Survation on behalf of Mark Diffley Consultancy and Research – found 63% were in favour of extending the period.
A UK-wide survey that ran alongside the poll found 52% of Britons against delaying the transition date.
Prime Minister Boris Johnson has continually insisted that the UK would reject any EU offer to extend the Brexit transition period beyond the end of the year.
The surveys were commissioned by a new campaign group, Starting up in Scotland.
Acting director of the campaign, Fergus Mutch, said: “The next few days are crucial. The sensible and practical thing to do would be for the UK Government to agree an extension to the transition period beyond the end of the year.
“Failing to do so leaves open the very real risk of a No Deal outcome, and significant damage to our future relationship with the EU.”
He added: “In the next few weeks we’ll launch an exciting new campaign to join the EU.
“This will be a grassroots movement with people at its core.”
The Scottish Government’s Constitution Secretary Michael Russell said the poll “reflects the continuing worry about Brexit and its disastrous consequences”.
He added: “Regardless of your view on Brexit, it’s painfully apparent that now is not the time to drag us over the cliff edge to a potential No Deal Brexit.
“The global economy is facing an unprecedented crisis and the last thing we need is to pile on yet more damage by ending the transition period this year.”
A UK government spokeswoman said: “The UK has left the EU, and the transition period ends on 31 December.
“This Government was elected with a clear manifesto commitment not to extend the transition and this position was endorsed by Parliament and reconfirmed by ministers in discussion with the devolved administrations.
“We are working hard to secure a deal with the EU which delivers for Scotland and the whole of the UK, so that Scottish businesses can take advantage of the huge opportunities outside the EU.
“And at the end of the transition period we will become an independent coastal state and take back control of our fishing water.”