The Liberal senator Dean Smith has said the parliament should consider a vote on a same-sex marriage bill this term, in contrast to conservative colleagues’ view that no vote should be held without a plebiscite.

Smith, who opposes the government’s proposed plebiscite, is the first Coalition MP to call for a parliamentary vote, which Malcolm Turnbull has not ruled out if the Senate blocks the plebiscite bill as is expected.

A range of Coalition conservatives including Josh Frydenberg, George Christensen and Andrew Broad have said the government should stick with the plebiscite policy, even if it is voted down in the Senate.

On Monday Smith told Sky News the government’s and prime minister’s position was that “the government’s path to marriage [equality] is via a plebiscite, full stop”.

“But the government is a subset of the parliament, the parliament is not a subset of the government,” he added.

Smith said if parliament tested the issue again it could pass same-sex marriage through a private members’ bill, as occurred on the abortion drug RU486.

There are now two same-sex marriage bills before the lower house, one moved by Labor and one by a crossbench group.

But as these are private members’ bills they may never be voted on if the government opposes them.

On Monday, Australian Marriage Equality and Australians For Equality launched a campaign to demand a free vote in parliament on same-sex marriage.

In August the same-sex marriage supporter and Liberal MP Trent Zimmerman warned there was no “plan B” if the plebiscite were blocked, and Coalition MPs would not have a free vote on a procedural motion to force a vote on marriage equality.

Asked how likely it was that parliament would change the Marriage Act without the prime minister’s agreement, Smith noted that the distinction between his position and Coalition colleagues who wanted a plebiscite “is not that clear”.

“Everyone agrees with a parliamentary vote where there is the point of difference is how you get to a parliamentary vote” and whether a plebiscite should be held first, he said.

“People will have different views, some people, I think, will be wanting to delay the issue [if the plebiscite is blocked].

“[If] pressed, I think it is in the government’s interests to have the issue dealt with one way or another quickly.”

Smith said most Coalition MPs were focused on the plebiscite bill, but bringing a same-sex marriage bill would address the concerns of a “large group [of Australians] in the middle that are fatigued by the issue ... [who] want it to be dealt with”.

“I think it’s defensible that at least once in the life of this parliament the issue should be tested, but not at the expense of the government’s priorities.”

Smith said he would abstain on the plebiscite bill, but if it came down to one vote he “would not be able to support [it]”.

He said plebiscites were not a feature of the constitution and had not been used since the early 1900s, with the exception of a vote on the national anthem.

“It is the role of parliament, we’ve done it on other contentious issues, like [abortion drug] RU486, divorce law reform. This is not the time and not the issue to experiment with plebiscites.”

Coalition MPs in favour of same-sex marriage who support a plebiscite, including Tim Wilson and Trevor Evans, have not said whether they support a parliamentary vote on marriage equality if the plebiscite bill is defeated.

Before Labor decided to block the plebiscite, same-sex marriage supporters in the Coalition warned that to do so could delay the reform by years.

On Monday, Wilson told Guardian Australia the plebiscite bill had not been considered by the Senate.

“Labor can still propose amendments so we can deal with it, get it through, change the law and the country can move on,” he said.

Co-chair of Australians For Equality, Anna Brown, said its new campaign would “build on the momentum and energy generated across the nation for marriage equality and continue to campaign for a straightforward change to be achieved by a vote in the parliament”.