It comes as immigration minister Robert Jenrick warned his colleagues to choose their words “carefully” as anger continued to grow over Ms Braverman’s “invasion” comments.
In a fiery Commons statement on Monday, Ms Braverman denied ignoring legal advice amid warnings that a temporary holding centre at Manston in Kent was dangerously overcrowded.
However, she has now facing calls to retract her comments, which have been labelled “"inflammatory" and "totally unhelpful" by opposition MPs and activists.
Mr Sunak told his Cabinet the UK would always be a “compassionate, welcoming country” in the wake of the home secretary’s comments. It is understood Ms Braverman’s comment to MPs – that “the British people deserve to know which party is serious about stopping the invasion on our southern coast, and which party is not” – had not been cleared with No. 10.
Questioned about Ms Braverman’s comments, Mr Jenrick told the BBC: “[Invasion] is not a phrase that I have used, but I do understand the need to be straightforward with the general public about the challenge that we, as ministers, face.”
On Sky News, Mr Jenrick claimed Ms Braverman had used the word “invasion” to describe the scale of the challenge.
He insisted: “In a job like mine, you have to choose your words very carefully.
"And I would never demonise people coming to this country in pursuit of a better life. I understand and appreciate our obligation to refugees. The scale of the challenge we’re facing is very, very significant.”
Around 40,000 people have crossed the English Channel in small boats so far this year. Mr Jenrick acknowledged the number could reach 50,000, with officials warning it could go even higher. The minister claimed his Cabinet colleague meant the issue of people coming here, not migrants specifically.
He said: “Invasion is a way of describing the sheer scale of the challenge. That’s what Suella Braverman was trying to express. She was also speaking, I think – and this is an important point – for those people who live on the south coast, who day in, day out, are seeing migrant boats landing on their beaches.”
Echoing Mr Jenrick, the Prime Minister’s official spokesman suggested Ms Braverman was seeking to “express the sheer scale of the challenge” at hand, but would not say if No. 10 would describe the situation in the same way.
The PM’s spokesman said he had not asked Mr Sunak how he would personally describe the scale of the issue, or whether it was inappropriate for the home secretary to use the word “invasion”.
Lord Dubs, who came to the UK as a child refugee in 1939, told Times Radio that Ms Braverman’s comments were "inflammatory" and "totally unhelpful".
He said: “It [language] is very important because it influences public opinion. It influences the way people see refugees and it shows hostility to people who are fleeing for safety. I just think a home secretary should know better than to use this language – the more so as she's been criticised by one of her junior ministers, Robert Jenrick, who said she has to be careful about the language one uses.
"So it's inflammatory, it's totally unhelpful and she shouldn't have said it.”
Speaking just days after a migrant centre was attacked, Lord Dubs claimed the UK was "becoming a less humane society".
He said: “We've always been seen as a country that upholds decent standards in human rights, and treats people in a humane and civilised manner. But can I just add this, that far more people claim asylum in Germany, in France and Spain and so on than they do here? So even the argument that they are all coming from France here is not true. Most of the asylum seekers actually stay in France and claim asylum."
Counter-terror police are meanwhile leading the investigation into the firebombing of the immigration processing centre in Dover which detectives suspect was sparked by “some form of hate filled grievance”.
Kent Police said the man suspected of carrying out Sunday’s attack was 66-year-old Andrew Leak from High Wycombe in Buckinghamshire.
After throwing two or three “crude” incendiary devices, Leak is believed to have killed himself at a nearby petrol station, police said.
Two members of staff at the centre suffered minor injuries. The suspect is believed to have posted anti-Muslim rants on Facebook.
Chief Inspector of Prisons Charlie Taylor said the Home Office needed to “get a grip” of the conditions.
He told Sky News: “What’s happening at Manston, when I visited, was people were sleeping on the floors, on the rubber mats down on the floors, and then very thin blankets or mattresses. [There were] lots and lots of people in a room, all squished in together, very uncomfortable.
“The room for families has lots and lots of different families all sharing the same room, very young children, older children. For a few hours, that would be acceptable, but where people are spending long periods of time there, it just isn’t.”
Shadow home secretary Yvette Cooper accused Ms Braverman of ramping up her rhetoric because she had no answers to the problems. “No home secretary serious about public safety or national security would use the language Suella Braverman did the day after a petrol bomb attack on a Dover centre,” Ms Cooper said. “But that’s the point. She isn’t serious about any of those things.”