Mr Abbott's sister, City of Sydney councillor Christine Forster, has hailed the results, saying she hopes it gives nervous Liberal MPs the confidence to vote with their consciences.

The Crosby Textor poll has found that support for same-sex marriage could switch votes to the federal Coalition, with 78 per cent of swing voters and people who have moved to or from the Coalition since the last federal election, supporting same-sex marriage.

''These are the people the Coalition really need to retain or regain,'' Crosby Textor director Jim Reed said. He added that the poll, which surveyed 1000 people for Australian Marriage Equality, also showed there were strong indications that Liberal support for same-sex marriage would be a vote winner, rather than a loser.

Sixty per cent of swing voters reported that it was an important issue for them and so might play a role in their vote choice. This compares with 74 per cent of those opposed to same-sex marriage, who said it wasn't important to them.

Libertarian NSW senator David Leyonhjelm is preparing a private member's bill to legalise same-sex marriage, but has said he will not bring it to a vote until it has a good chance of success.

Ms Forster, who is engaged to marry her partner Virginia Edwards once same-sex marriage is legal either federally or in NSW, described the poll as ''really positive'' news. ''I would hope that the polling results would give members of the Liberal Party, who might otherwise feel a bit nervous about how the electorate view this issue, some confidence,'' she said.

The Liberal Party is yet to confirm its position on same-sex marriage in the new Parliament. While Mr Abbott's firm position is that marriage should be between a man and a woman, he has said the question of whether there is a free vote on the issue within the Liberal Party would be a matter for the party room.

In order to bring on a formal discussion in the party room, a bill needs to be before the Parliament.

Senator Leyonhjelm is still finalising his bill, but Liberal sources say it will be a lot more palatable to Liberal MPs because he is arguing same-sex marriage is not a left-wing issue.

''It's a liberty issue,'' the Liberal Democratic Party senator said, adding that his message to MPs is: ''You don't have to approve, you just have to tolerate.''

Australian Marriage Equality national director Rodney Croome said in light of the polling, allowing a free vote in the Liberal Party on same-sex marriage was ''not only the right thing for the Coalition to do [but] also the smart thing''.

''Removing obstacles to marriage equality is a sure-fire way for the federal government to reclaim some of the support it has lost in recent months,'' he said.

Mr Croome said a vote in support of same-sex marriage would be close at the moment in the Senate. But support for same-sex marriage is about 20 short of a majority in the House of Representatives.