The home secretary said she was “disappointed” by the Match of the Day host’s comments.
Lineker, who has 8.6 million Twitter followers, commented on a video on the social media site put out by Braverman, in which she unveiled government plans to stop migrant boats crossing the Channel.
He tweeted : “Good heavens, this is beyond awful.”
The former England striker then replied to a user who challenged him because it was “easy to pontificate when it doesn’t affect you”.
Lineker wrote: “There is no huge influx. We take far fewer refugees than other major European countries.
“This is just an immeasurably cruel policy directed at the most vulnerable people in language that is not dissimilar to that used by Germany in the 30s, and I’m out of order?”
Lineker will be “reminded of his responsibilities on social media” by the corporation following the tweet reply, according to the Daily Telegraph.
Asked about his remarks on BBC Radio Four’s Today programme, Braverman said: “I’m obviously disappointed that he should attempt to equate our measures with 1930s Germany. I don’t think that’s an appropriate way of framing the debate.
“As I’ve made clear, we are working very hard to ensure that people stop dying on the channel - that’s what this is about. People are taking this treacherous journey in the false hope that this will lead to a life in the United Kingdom.
“There are many other routes that they can use to come here and that’s what I encourage them to do.”