A BBC spokesperson would not comment.
Mr Thaker retweeted the claim that Newsnight presenter Ms Barnett was “using the same old antisemitism excuse whenever people criticise Israel” when she spoke about her view of the tweets sent by the antisemitic rapper Wiley.
He also shared claims that Israel was “a racist state” and regularly defended Jeremy Corbyn over his failure over antisemitism.
Mr Thaker also described Tim Davie, the new BBC director-general, as a “white male Tory.” Culture Secretary Oliver Dowden had described the JC’s report on Mr Thaker’s activity as “very concerning.”
A BBC source said the case had caused much concern within the corporation - particularly the tweets directed at Ms Barnett, who is widely respected.