The Top Gear presenter was the subject of an internal investigation at the corporation after he attacked producer Oisin Tymon.
Mr Tymon was left with a split lip and received a torrent of verbal abuse during the 30-second assault on 4 March.
Clarkson was suspended by the BBC on 10 March before it was announced that his contract would not be renewed.
Clarkson will record his appearance on Have I Got News For You, on which he has appeared before, on 23 April and the show will be broadcast the day after.
A BBC spokeswoman said: “Jeremy’s contract has not been renewed on Top Gear but he isn’t banned from appearing on the BBC.”
Jimmy Mulville from production company Hat Trick, which makes the show, told the Broadcasting Press Guild last month that he thought Clarkson was “a fantastic broadcaster”.
He joked: “Maybe we will get the producer on so he can hit Jeremy Clarkson live on television.”
Clarkson’s wider future remains unclear, although North Yorkshire Police announced on Tuesday he will not face police action over his attack on Mr Tymon, which happened at a hotel in the area.
It emerged last week that he will appear on stage with his former Top Gear colleagues, Richard Hammond and James May, later this year to fulfil a series of shows planned before he left the BBC.
The gigs will be stripped of all BBC branding and content, and billed as Clarkson, Hammond and May Live.
The move has sparked rumours that the three men intend to team up to continue their careers together after Clarkson’s exit from the corporation, but sources close to the tour say it is just a matter of fulfilling commitments to the fans.
The future of the show’s other two presenters – Hammond and May – is unclear, with both men’s Top Gear contracts having run out.