An Ipsos Mori survey found 68% of 1,003 people asked agreed that religious organisations should be able to marry same-sex couples if they want to.
The survey was carried out for the Equality Network, LGBT Youth Scotland and the Scottish Youth Parliament.
But a church spokesman said asking whether people support the right to do something solicits a positive response.
The survey results, which also indicated 64% support for the right of same-sex partners to marry, come as the Scottish government is due to publish the results of its consultation on the issue this month.
Consultation process
The government launched the consultation stating it "tends towards the view" that same-sex marriage should be introduced but that faith groups and their celebrants should not be obliged to solemnise the ceremonies.
The Scotland for Marriage group, which is supported by some religious groups, has campaigned against the proposal.
But the Faith in Marriage campaign - a coalition including, among others, the United Reformed Church, the Quakers, Buddhists and the Pagan Federation - has spoken out in support.
Speaking to the BBC's Sunday Politics Scotland programme, Peter Kearney, spokesman for the Roman Catholic Church in Scotland, said the poll was "deeply flawed".
He said: "The public in general are very wary and unlikely to suggest a person shouldn't have a right. When you begin your question by saying should someone have the right to same-sex marriage you automatically have distorted the result. To be honest the poll that matters is the three-month-long consultation that took place at the end of last year and in that one we know that the vast majority said marriage shouldn't be redefined."
Tom French, from the Equality Network and Equal Marriage Campaign, said: "Around the world increasing majorities of the public are saying that they support same-sex marriage. It's supported in parliament as well, last week we announced that a majority of MSPs have signed our equal marriage pledge saying that they will vote for this. So we're saying to the Scottish government - listen to the public, listen to the parliament and you've got the green light to go ahead with legislation."
Mr French said such legislation would not force religious organisations to conduct same-sex marriages if they opposed it.
But Mr Kearney said that any opt-outs being proposed simply cannot be delivered under UK and European legislative frameworks.