Boris Johnson’s government is facing demands to give parliament greater powers to scrutinise and approve the UK’s international trade deals as the prime minister prepares to unveil the first significant post-Brexit agreement on Tuesday with Australia.

Writing to Liz Truss, international trade secretary, a cross-party group of more than 20 MPs said they had “urgent concerns” over the impact of the pact with Canberra, which has split the cabinet and angered farmers’ groups.

Under UK law, trade deals are signed before parliament has the chance to debate their contents, leading trade groups and MPs to warn that the government risks setting precedents for future trade policy without proper debate.

The MPs warned Truss that effective scrutiny was essential in order to allay fears that British farmers will struggle to compete against Australia’s “farming industrial complex”.

“Any trade deal agreed with Australia must receive proper scrutiny and approval by parliament to assuage our concerns and the concerns of the public,” they wrote in the letter, co-ordinated by pro-internationalist campaign group Best for Britain.

An announcement that an “agreement-in-principle” has been reached on the deal is set to be made on Tuesday. Boris Johnson met Scott Morrison, his Australian counterpart, on Monday in Downing Street and reached a tentative accord.

The 2010 Constitutional Reform and Governance Act means the British parliament gets to scrutinise international treaties, including trade deals, only once the ink is on the paper.

The Johnson government has successfully fought off attempts by MPs to amend the law to give them greater oversight powers before that point, according to Brigid Fowler, senior researcher at the Hansard Society, a think-tank.

It did agree some legally non-binding concessions, such as potentially holding debates on UK negotiating objectives, but will still allow formal scrutiny only after the deal is complete.

“What we have so far is better than pre-Brexit default practice but still a considerable distance from the strongest parliamentary roles internationally,” Fowler said.

One senior Department for International Trade insider said the trade agreement would receive “full parliamentary scrutiny” and MPs would be free to block a deal if they chose to.

“Our scrutiny arrangements are some of the most robust and transparent in the world, and the Trade and Agriculture Commission [a governmental advisory body] will play a full role, providing expert and independent advice. The next stage is only the deal being agreed in principle. The full signing won’t take place until months afterwards when the scrutiny process really kicks in,” the official said.

Sir Roger Gale, a Conservative party MP, said the public was united on the need to not expose the UK to products made under lower standards. “We need to hold the government to account and ensure the promises made to the electorate are kept.”

Hilary Benn, the Labour MP who formerly chaired the Brexit select committee, said the Australia deal risked setting “a significant precedent” for future deals with larger markets.

He added that recent rows over the implementation of the Northern Ireland protocol had highlighted the risks of rushing through international treaties “for short-term political gain while failing to understand the long-term consequences”.

Chris Southworth, secretary-general of the International Chamber of Commerce UK, said the government needed to accept more public and parliamentary scrutiny in order to win better deals than those it had negotiated thus far. “We are significantly behind other governments like the US, Chile or Canada that have more robust, transparent systems,” he said.

Nick von Westenholz, director of EU exit and international trade at the National Farmers’ Union, said the current system of scrutiny was “seriously underpowered” and risked losing public support if MPs were not given more time for review and debate.

“One thing the referendum taught us is that the public is fed up with important decisions being made by faceless bureaucrats rather than elected MPs. But when it comes to trade deals, we are being presented with a fait accompli,” he said.

One ally of Truss said: “She will never agree to any deal that undermines our high standards, and any deal we do sign will include protections for farmers. The Australia deal is an important gateway into the Trans-Pacific Partnership, which will open massive new opportunities for farmers and deepen access to consumer markets of over half a billion people.”