“Good heavens, this is beyond awful,” he wrote.
Gary then responded to one Twitter user who claimed it was “easy” for the football pundit “to pontificate when it doesn’t affect you”.
He wrote back: “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?”
Shortly after his posts, the Telegraph quoted a “BBC source” who said that Gary would be “reminded of his responsibilities on social media” by the corporation.
A BBC source also told the PA news agency that the BBC is taking the matter “seriously” and expects to have a “frank conversation” with Gary about his tweets.
Meanwhile, a BBC spokesperson said: “The BBC has social media guidance, which is published. Individuals who work for us are aware of their responsibilities relating to social media.
“We have appropriate internal processes in place if required.”
The BBC had no additional comment when contacted by HuffPost UK.
Gary’s comments have already been widely condemned by Tory politicians, who urged the BBC to take action.
During Wednesday’s edition of Radio Four’s Today programme, Suella Braverman said of Gary’s remarks: “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.”