The remarks, which came amid a separate row within the Tory party about record levels of net migration to the UK, raised eyebrows among some in the party.
Mr Cleverly insisted, in an interview with The Times, that the initiative is not the “be all and end all” to stopping Channel crossings.
“My frustration is that we have allowed the narrative to be created that this was the be all and end all,” he said.
Mr Sunak, in contrast, used an interview with The Mail On Sunday to stress the importance of the scheme, after the Supreme Court ruled it unlawful earlier this month.
Speaking to the Sunday Morning With Trevor Phillips programme on Sky News, Ms Trott said: “They’re both actually saying the same thing, which is that Rwanda is part of our plan.
“Both saying it is part of the plan, it is not all of the plan.”
Mr Sunak has pledged not to let a “foreign court” stop flights to Rwanda, with plans for a new treaty and emergency legislation to ensure the plan is legally watertight.
It was the UK Supreme Court, rather than “a foreign court”, that dealt the latest blow to the Government’s hopes of sending asylum seekers who arrive in the UK on a one-way trip to Rwanda.
But Tories are keen to ensure that the ECHR and the Strasbourg court that rules on it will not prevent the policy, first announced in 2020, from being implemented.
Ms Trott said: “We have successfully in the last year bought the numbers of people coming over here illegally down by a third.
“That is at a time when the numbers coming into Europe are up by 80%.
“This was not a foregone conclusion.”
The Cabinet minister declined to spell out any new steps the Government might take to reduce overall net migration, another preoccupation of Tory MPs.
The figure peaked at 745,000 in the year to December 2022, according to revised estimates published by the Office for National Statistics (ONS) on Thursday.
The data places migration levels at three times higher than before Brexit.
Immigration minister Robert Jenrick is understood to have worked up a plan designed to appease calls from right-wing Tories for the Government to take action.
He is pushing for a ban on foreign social care workers from bringing in any dependants and a cap on the total number of NHS and social care visas.
His plan would also scrap the shortage occupation list, a programme that allows foreign workers to be paid 20% below the going rate in roles that suffer from a lack of skilled staff.
But Ms Trott, who said immigration levels are “too high”, declined to shed any light on what potential measures could be introduced.
“This year we brought forward a £600 million plan to train more people to do social care in this country.
“So we are taking concrete steps, I’m not just saying here I want it to come down, I’m saying that we are taking concrete steps to bring it down,” she told the BBC’s Sunday With Laura Kuenssberg programme. By Dominic McGrath, The Telegraph