Meanwhile, Liberal MPs who wanted to bring same-sex marriage to a head in the next fortnight of Parliament say their plans have hit a wall, and now concede movement on the issue is unlikely in this half of the year.

The letter from 20 of the country's chief executives calling on the Turnbull government to legislate for same-sex marriage appears to have backfired, with moderate Liberal MPs labelling it "unhelpful".

In the third successive day of his crusade, Mr Dutton lashed out at social media activists who were "blackmailing" companies by threatening boycotts, and warned the government would not abandon its policy of holding a plebiscite.

A crowded agenda in the lead-up to the budget, including energy policy and reforms to racial discrimination law, have also slowed momentum for change in the minds of moderate Liberals.

"There is no pathway at this stage," one supportive MP told Fairfax Media. "We're constantly trying to find a way. There isn't the momentum to convince the party room."

But the issue has not been wiped off the agenda completely, with MPs refusing to rule out a party room ambush or even crossing the floor of Parliament in the second sitting week of the next fortnight.

Such actions were being "actively considered" and remained "unlikely but not impossible", one MP said. There was also division among supporters about the best way of bringing the issue to a head.

One option that has fallen by the wayside is a cross-party bill originating in the Senate, based on the government's draft Marriage Act amendments, which could then force a debate in the party room.

Drafting of a cross-party bill was mooted and investigated weeks ago, but has not been commissioned, and sources from both major parties now argue any action would need to start in the lower house.

In February, Fairfax Media named Liberal MPs Warren Entsch, Trent Zimmerman, Trevor Evans, Tim Wilson and Dean Smith as a group of MPs pushing the government to abandon its plebiscite policy.

Mr Dutton on Saturday ramped up his attack on business leaders who are also agitating for change, warning the Turnbull government was "not going to be bullied" by chief executives "or anyone else".

"It is unacceptable that people would use companies and shareholders' money of publicly listed companies to throw their weight around," Mr Dutton said.

"If Alan Joyce and any other CEO wants to campaign on this or any other issue in their own time and on their own dime, good luck to them.

"If you want to run for politics, run at the next election and have your say. We don't want multibillion dollar companies with all their resources weighing into social debates. That's my call."

Mr Joyce was travelling on Saturday afternoon but in a statement Qantas defended its participation in the campaign, and vowed to continue to "speak up" on issues it deemed important.

"Qantas engages on a number of social issues, from Indigenous reconciliation to gender diversity and marriage equality," a spokesman said.

"We do so because we believe these issues are about the fundamental Australian value of fairness, and we're the national carrier."

Mr Dutton also stepped up his crusade against the social media backlash to Coopers' involvement in a promotional video featuring a debate on marriage equality between Mr Wilson and Andrew Hastie.

Companies were being "morally coerced" into supporting same-sex marriage out of fear they would be boycotted and "extorted" by online activists, he said.

"This is a battle for common sense and for freedom of speech, make no mistake about it. It is part of a culture war that is raging in our country at this very day," Mr Dutton told a Liberal National Party conference in Cairns.

"We cannot accept this politically correct notion that somehow it is OK for online bullying and for online extortion of companies, blackmailing them into submission."

Mr Turnbull on Friday reiterated the government's long-held position of holding a public vote on marriage equality.

"We have a commitment on a plebiscite which we took to the election, so that's our policy," he said.