Marriage equality should be settled in this term of parliament, if possible, the industry minister, Arthur Sinodinos, has said.

Sinodinos made the comments at the National Press Club on Wednesday, after delivering a speech about the need to harness digital disruption and reinsert scientific facts into public policy debate.

Pressure is growing on the Turnbull government to deal with same-sex marriage before the next election, with the immigration minister, Peter Dutton, proposing a voluntary postal plebiscite and the education minister, Simon Birmingham, suggesting the issue could return to the Liberal party room.

This week the Liberal moderate MPs Trent Zimmerman and Warren Entsch publicly called for a free vote, joining senator Dean Smith, who has been calling for that since last year.

“I think it would be good for that to be settled, if possible, in this parliament,” Sinodinos said when asked about same-sex marriage.

He said the challenge was that the government had proposed a plebiscite at the election, meaning it now claimed a mandate for the policy but the Senate claimed a mandate to block it.

“Fair enough. My point is this: I don’t want people coming back to me and saying, ‘Hang on, we want you to break a promise’, because that is what we are being told to do.

“So the dilemma for the Coalition is we went to the election with a certain policy.

“At the moment, we are still committed to that policy ... in the sense there are no plans to change it and therefore I can’t see we can do anything but prosecute the case to stay with that approach.”

On ABC Adelaide on Wednesday, Birmingham said he hoped the parliament would support a plebiscite bill to “ensure all Australians can have their say”.

“If that fails then, of course, we can always have discussions in the Liberal party room about how such matters could be addressed in the future.”

On Tuesday Fairfax Media reported that Dutton and Mathias Cormann back a voluntary postal plebiscite on marriage equality as a means to fulfil the government’s promise but resolve the issue before the next election.

In response, Zimmerman called for a free vote in parliament and argued a postal plebiscite would appear “tricky and sneaky”. He warned the postal vote would be non-binding and its result could be disregarded and it would result in a drawn-out three-month campaign.

On Wednesday the Liberal MP and marriage equality advocate Warren Entsch told Guardian Australia: “I commend [Dutton] for looking at options. I welcome the acknowledgement that something needs to be done.

“I’ll work with him. To me, it’s not a matter of should we or shouldn’t we [deal with marriage equality], it’s a question of how.

“People don’t have to change their views about whether they support it or not, it’s how we develop [a process to resolve it].”

Entsch said his own position was clear: “I would rather have had a [free] vote a long time ago ... That has always been my position.”

Turnbull has never ruled out a free vote but reiterated his support for a plebiscite as recently as Friday.

Asked about a possible postal plebiscite, a government spokesman told Guardian Australia the prime minister would not comment on “rumours and speculation”.

“Our policy on marriage equality is clear,” the spokesman said. “The Australian people should get a say on this important issue.”

Asked if this left the door open to a postal plebiscite, the spokesman added: “The policy is for a plebiscite not a postal plebiscite.”

Australians for Equality director, Tiernan Brady, said the postal plebiscite was “not a serious proposal ... [that would] deliver marriage equality in line with the will of the majority of the Australian people”.

Brady said the proposal would be seen as sneaky because it bypassed parliament, which had rejected the plebiscite.

“It’s parliament’s job to pass laws that protect Australians, so why should only one group be subjected to this special process?”

On Monday the Liberal MP Craig Kelly told Guardian Australia the postal plebiscite would allow a popular vote at a “fraction of the cost” of the proposed $170m compulsory plebiscite.

“The [plebiscite] bill is blocked in the Senate and the issue is going nowhere,” he said. “There’s pressure on the government to bring [same-sex marriage] back to the parliament and break an election commitment.

“So the question is, how can you try again? I think the best solution is to deal with the issue ... by going for a direct vote.”

Kelly said it was important for the issue to be dealt with by a popular vote because it would increase acceptance of same-sex marriage.