The new treaty, which would then take at least three weeks to be approved by British lawmakers, follows a Supreme Court ruling on Wednesday that blocked the current deal between the two countries from proceeding.
Wednesday's ruling witnessed a five-judge panel unanimously siding with an earlier Court of Appeal decision that the policy to deport asylum seekers and refugees to Rwanda was incompatible with Britain's international obligations.
The judges agreed that the East African nation was not a safe third country and there were "substantial grounds" to believe it could forcibly return migrants to places where they could face persecution.
The Supreme Court ruling was a major setback for Prime Minister Rishi Sunak, who is under pressure to reduce irregular immigration to Britain.
Sunak immediately vowed to persevere with the controversial plan by securing a new treaty with Kigali.
The British prime minister says he plans to introduce "emergency legislation" to parliament to designate Rwanda a safe country to end the "merry-go-round" of legal challenges.
Speaking during an interview with Sky News on Thursday, James Cleverly, Britain's Interior Minister said, "it is ready pretty much now to turn into a treaty that can be done within days —not weeks or months."
He added the legally binding treaty would stipulate Kigali cannot deport migrants sent from Britain onto other countries which could be deemed dangerous, in a bid to negate the legal concerns.
"We've been told by the judges what it is we need to fix in order to get these flights off the ground — we are fixing those things," Cleverly said in another interview with Times Radio.
British authorities insist that the asylum deal with Rwanda is crucial to deter "illegal" immigration across the Channel from France on inflatable vessels — an emotive issue set to feature prominently in the next general election.
The Supreme Court decision exposed rifts in the ruling Conservative party over the issue and is set to prompt a renewed push from right-wingers for withdrawal from the European Court of Human Rights, ECHR.
The calls by the right-wingers came after a last-minute injunction by the ECHR in June 2022, stopping the first plane that was scheduled to carry deportees to Rwanda, prompting the legal challenges.
Sunak said Wednesday that if the court "chooses to intervene against the express wishes of parliament, I am prepared to do what is necessary to get the flights off."
But in his interviews, Cleverly noted that Britain's "preferred option" was to remain in the ECHR.
"We don't think we're going to need to (leave)," he said.