Labor will put forward a private member’s bill for marriage equality in the new parliament, the shadow attorney general, Mark Dreyfus, has revealed.

In an opinion piece for Guardian Australia published on Wednesday, Dreyfus promised that Labor would put forward a bill in the first weeks when parliament returns on 30 August.

A private member’s bill could legislate same-sex marriage without a plebiscite but is unlikely to succeed given the slim Coalition majority in the lower house.

However, the bill may pass the Senate, where the Coalition does not have a majority, and could put further pressure on the prime minister, Malcolm Turnbull, to allow a free vote on marriage equality.

Dreyfus criticised the plebiscite as an invention of the then prime minister, Tony Abbott, “as a delaying tactic and a way to shore up his teetering leadership”.

He notes that, since the 2 July election, conservatives in the Liberal party have called for the plebiscite to be delayed, making it a “ticking time bomb” for the re-elected Turnbull government.

Dreyfus wrote Turnbull had “self-interest in avoiding the plebiscite” as it “could be the issue which brings down his leadership”. He called the prospect of delay “a disgusting way to treat the hopes and dreams of LGBTI Australians”.

“Labor believes there is enough support in the parliament if a conscience vote were to take place to make marriage equality a reality within the first few weeks of parliament,” he wrote. “We call on Mr Turnbull to show leadership and consider allowing this to happen.”

Dreyfus said a free parliamentary vote would “avoid a hurtful public debate and it would show that Mr Turnbull is not afraid of his own backbench”.

“It would give LGBTI Australians the equal rights they deserve, bring Australia into the 21st century and achieve marriage equality without a vitriolic public debate.”

But Dreyfus said he had “very strong doubts” Turnbull would allow the vote, saying it was “likely he will fail” in the test of leadership.

Despite arguing against a plebiscite in the party room in August, since becoming prime minister in September Turnbull has retained the policy.

Turnbull has promised the plebiscite would be held as soon as practicable, although he conceded that might not be until 2017.

The government has claimed a mandate for the plebiscite since being re-elected on 2 July, despite not having released the question or finalising details including whether the for and against cases will receive public funding.

Even key supporters of same-sex marriage in the Coalition – including the MPs Warren Entsch, Trent Zimmerman and Tim Wilson – now support a plebiscite as the way forward.

But, as recently as March, Zimmerman promised to cross the floor and support a substantive bill on marriage equality.

A spokesman for Zimmerman told Guardian Australia the plebiscite was the best option to achieve same-sex marriage and “he hopes that Labor and the Greens see this and support the [plebiscite enabling] legislation”.

“His understanding is that a private member’s bill is not on the government agenda to be debated.”

On Sunday the attorney general, George Brandis, said cabinet would consider the plebiscite enabling legislation in coming weeks and revealed it would allow electorate by electorate reporting of results.

Dreyfus cricitised the move, saying the only reason to allow electorate by electorate reporting was “to curry favour with opponents of marriage equality, to give them an ‘out’ from following the national vote”.

Before the 2 July election Labor promised to bring a marriage equality bill within 100 days if elected. However, this is the first time the opposition has confirmed it would bring a private member’s bill after the leader, Bill Shorten, conceded defeat in the election.

Labor’s new shadow assistant minister for equality, Terri Butler, who sponsored a cross-party marriage equality bill in 2013, has previously said she would “do whatever I can do to avoid a plebiscite”.