A chartered plane is set to leave one of London's airports overnight and land in Kigali on Wednesday, campaigners said, after British judges on Monday rejected an appeal against the deportations.
Claimants had argued that a decision on the policy should have waited until a full court hearing on the legality of the policy next month.
Thirty-one migrants were due to be sent but one of the claimants, the NGO Care4Calais, tweeted that 23 of them had now had their tickets cancelled.
Those due to be deported include Albanians, Iraqis, Iranians and a Syrian, Care4Calais said.
Other claimants included the Public and Commercial Services Union (PCS), whose members will have to implement the removals, and immigration support group Detention Action.
PCS chief Mark Serwotka has said it would be "an appalling situation" if Tuesday's removals were subsequently found to be illegal at the full hearing in July.
Home Secretary Priti Patel should wait until then if she "had any respect, not just for the desperate people who come to this country, but for the workers she employs", Serwotka told Sky News.
Protesters gathered outside the Royal Courts of Justice and the Home Office on Monday.
In Geneva, UN refugee chief Filippo Grandi has denounced the UK government policy as "all wrong" and said it should not be "exporting its responsibility to another country".
Church of England leaders, including top cleric the Archbishop of Canterbury, Justin Welby, stepped up criticism of the policy as "one that should shame us as a nation".
"Our Christian heritage should inspire us to treat asylum seekers with compassion, fairness and justice, as we have for centuries," Welby and 24 other bishops wrote in Tuesday's Times newspaper.
"This immoral policy shames Britain."
"Evil trafficking" must be combatted by providing safe routes to the UK to "reduce dangerous journeys", The Times quoted the bishops as saying ahead of the letter's publication.
The British government insists the policy is needed to stop a flood of all-too-often deadly migrant crossings of the Channel from France.
"I can't say exactly how many people will be on the flight," Foreign Secretary Liz Truss told Sky News on Tuesday.
"The really important thing is that we establish the principle and we start to break the... business model of these appalling people, traffickers who are trading in misery," she added, reiterating remarks made by Prime Minister Boris Johnson on Monday.
"It's very important that the criminal gangs who are putting people's lives at risk in the Channel understand that their business model is going to be broken," Johnson told LBC radio.
"They're selling people falsely, luring them into something that is extremely risky and criminal."
Under the agreement with Kigali, anyone landing in the UK illegally is liable to be given a one-way ticket for processing and resettlement in Rwanda.
The UK government says that genuine asylum claimants should be content to stay in France.
Contradicting the UN refugee agency UNHCR, it insists that Rwanda is a safe destination with the capacity to absorb possibly tens of thousands of UK-bound claimants in the future.
Doris Uwicyeza, chief technical adviser to Rwanda's justice ministry, pushed back against criticism of the human rights record of President Paul Kagame's government -- which is set this month to host a Commonwealth summit attended by Prince Charles and Johnson.
Rwanda's 1994 genocide made it particularly attentive to "protecting anybody from hate speech and discrimination", including gay people, she told LBC radio.
British newspapers reported that Prince Charles had dubbed the plan "appalling".
The reported comment prompted unnamed cabinet ministers to tell Queen Elizabeth II's heir to stay out of politics.
International NGO Human Rights Watch issued a public letter warning that "serious human rights abuses continue to occur in Rwanda, including repression of free speech, arbitrary detention, ill-treatment, and torture".