The Match Of The Day host criticised the proposed crackdown, appearing to compare it to Nazi Germany in a series of posts on Twitter on Tuesday.
Shortly afterwards, the Telegraph quoted a “BBC source” who said that Gary would be “reminded of his responsibilities on social media” by the corporation, which has strict impartiality rules.
However, Gary remained defiant in his criticism of the government’s plans, and in a tweet on Wednesday, thanked followers for supporting him in doing so.
“I have never known such love and support in my life than I’m getting this morning (England World Cup goals aside, possibly),” he said. “I want to thank each and every one of you. It means a lot.
“I’ll continue to try and speak up for those poor souls that have no voice. Cheers all.”
Gary had originally branded the government’s plans “beyond awful” and “immeasurably cruel”, writing: “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?”
His comments were condemned by a number of 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.”
Meanwhile, a BBC source has told the PA news agency that the corporation is taking the matter “seriously” and expects to have a “frank conversation” with Gary about his tweets.
A spokesperson for the broadcaster 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.”