There is little support inside the Abbott ministry to use a referendum to settle the question of same-sex marriage.
Attorney-General George Brandis has dismissed as "entirely unnecessary" the need to change the constitution, arguing it already gives the commonwealth parliament power to legislate marriage laws.
Other ministers have sided with Mr Brandis to distance themselves from Social Services Minister Scott Morrison who floated the idea.
Christopher Pyne told the Nine Network on Friday: "A referendum would cost a great deal of money, in fact only to achieve no outcome because there is no legal basis for a referendum."
Assistant Treasurer Josh Frydenberg backs a plebiscite - a non-binding popular vote - over a referendum.
"What George Brandis was saying was right; you do not need a referendum to change the constitution in order to enable the parliament to make a change in relation to gay marriage," he told ABC TV.
The High Court made that clear just two years ago, Mr Frydenberg said.
But he conceded Mr Morrison's point that a referendum produces a binding decision on parliament whereas a plebiscite was indicative of community attitudes.
The same-sex marriage supporter described this week's six-hour meeting of coalition MPs as a "cathartic" experience for the government.
But like cabinet minister Malcolm Turnbull, Mr Frydenberg ruled out crossing the floor to support a cross-party bill Liberal colleague Warren Entsch will introduce to parliament next week.
Opposition Leader Bill Shorten, who has his own private bill before parliament, remains dismissive of holding a referendum or plebiscite.
"If you want to see change in this country, then I think it starts with changing the prime minister," he told ABC radio.
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