The International Trade Secretary told MPs the deals agreed in principle with the two countries would soon be formalised. It comes after over a year of intense negotiations.
She told the Commons International Trade Committee: "We're just in the final throes."
An agreement with Australia was first announced in June, while a pact with New Zealand was secured last month.
Despite one being agreed four months before the other, the legal texts to formalise both deals are set to be signed at a similar time.
Ms Trevelyan said: "With Australia, we are further along on the process to the final completion of the legal text.
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It is hoped the agreements with help Britain in its plans to join the Comprehensive and Progressive Agreement for Trans-Pacific Partnership made up of 11 countries.
She said striking bilateral deals with members already a part of the bloc would help speed up the process.
The minister said: "Where you have a bilateral arrangement already with a country that makes it much easier for them to say 'well, we already know this country', in a simplified way, 'we have a great relationship so we have no anxieties we're bringing in a country into the CPTPP family that wouldn't have the right perspectives and same framework on free trade."
The CPTPP was worth 110billion of UK trade in 2019.
While Ms Trevelyan has hailed the agreements as "fantastic" for the United Kingdom, the National Farmers' Union has criticised the amount of foreign meat that can be imported under the deals.
Brushing off the concerns, the Trade Secretary told the committee: "I think this is great for farmers and I think it is a fantastic deal for removing tariffs on all food and drink exports, from gin and chocolates, to pork and wine.
"There is a great range of liberalisation on all of those.
"We will include protections for our agriculture industry where there are sensitivities - a range of tools to defend British farmers against any unfair trading practices that could lurk, and those things like tariff liberalisation on sensitive goods like beef and lamb will be staged over time."
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Brexit Britain closing in on TWO mega trade deal: 'We're just in the final throes' - Daily Express
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