The row comes as Ms Braverman fights in cabinet for the government to go further in curbing net migration.
Figures to be released on Thursday are expected to show it hit an all-time high last year. The total is estimated to be between 700,000 and one million.
She has publicly pressured Mr Sunak, saying the government should “reduce overall immigration rates” and train British workers so that the UK “don’t forget how to do things for ourselves”.
It is the second time that questions about Ms Braverman’s behavior have come up during an argument over immigration.
While objecting to plans to let more migrants into the country last October, Ms Braverman was forced to resign from Liz Truss’ cabinet after it emerged she had sent an official document to a colleague from her personal email.
Miriam Cates, a Tory MP and ally of Ms Braverman, said it was “extremely concerning” that the Home Secretary was being subjected to personal attacks and “a concerted effort to discredit her”, in a similar way to last October.
“It is no coincidence that it was in the same week that she was very vocal about the need to put proper limits on legal migration, which is clearly a contentious issue in government,” Ms Cates said.
“She has had a lot of profile and fame. There are many people who disagree with her view that we should restrict legal immigration. If you add two and two together, it is quite possible that it is politically motivated. It is shocking that someone would leak this private information.”
Craig Mackinlay, another Tory MP, said: “It was not at all unreasonable for a Home Secretary to do what she asked. I don’t see anything wrong with that. I would call that reasonable common sense and then decide to take the points.”
There has been speculation in Whitehall that Ms Braverman would be willing to resign if she feels immigration is not being addressed. However, the allegations about her behavior would lessen the impact of such a move.