Deputy Prime Minister Barnaby Joyce wants his coalition partners to concentrate on hot-topic issues, such as power prices, rather than same-sex marriage.
The Nationals leader denies internal divisions resurfacing within the Liberal Party have the potential to threaten Malcolm Turnbull's prime ministership.
Mr Joyce's comment was backed by new research showing same-sex marriage has virtually disappeared as a political issue a year on from the election.
Cabinet minister Christopher Pyne ignited the row after telling moderate faction colleagues at a late-night federal Liberal council function last week the same-sex marriage issue could be resolved sooner than everyone thought.
"Your friends in Canberra are working on that outcome," he can be heard saying in a leaked recording.
Mr Joyce, who is in London for trade talks, played down the comments.
"What someone says late at night after a function is not really consequential to where our nation is going," he told ABC TV on Tuesday
"People want you to concentrate on the main game, which is for us getting a better return through the farm gate. For most people concentrating on issues like power prices."
Research by analysts Meltwater, provided to AAP on Tuesday, found social media and news coverage of the same-sex marriage issue peaked during last year's election campaign at 74 per cent "share of voice", dropping to only seven per cent between May 1 and June 20 this year.
While unions, school funding and marriage had been the highest trending social media terms relating to Malcolm Turnbull and the Liberals during the election campaign, this had turned around to schools, Medicare, terrorism and coal this year.
The prime minister told reporters in Melbourne on Tuesday the government remained committed to a plebiscite before MPs voted on the law change.
Liberal MPs Trent Zimmerman and Dean Smith are believed to be working on a private bill they hope will get same-sex marriage through parliament with a conscience vote.
Attorney-General George Brandis and Mr Pyne were aware of the bill, Fairfax Media said.
Mr Pyne sought to clarify his taped comments when he appeared on ABC-TV's Q&A program on Monday night, insisting he was merely speaking to supporters and telling them to have hope there will be marriage equality eventually after a public vote.
Conservative Liberal senator Eric Abetz, who staunchly opposes same-sex marriage, urged coalition MPs to stick with the plebiscite.
"I call on my colleagues - especially cabinet ministers - to ensure they faithfully support that which cabinet supports, that which their party room supports, that which we took to the election just one year ago," he told ABC radio.
Former prime minister Tony Abbott, who devised the same-sex marriage plebiscite plan the coalition took to the 2016 election, has warned dumping a poll would be a breach of faith with the people.