The star was the subject of an internal BBC investigation after he attacked Top Gear producer Oisin Tymon, splitting his lip and verbally abusing him in a 30-second assault on March 4.
He was suspended by the BBC on March 10 before it announced his contract would not be renewed.
Clarkson received widespread public support — including from his friend Prime Minister David Cameron — in the aftermath of the fracas, and 1m people signed a petition calling for the BBC to reinstate him.
However, director general Tony Hall said ”a line has been crossed” and ”there cannot be one rule for one and one rule for another”.
Clarkson will record his appearance on Have I Got News For You, on which he has appeared before, on April 23 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, said 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 have announced he will not face police action over his attack on Tymon, which happened at a hotel in the area.
It emerged last week that he will appear on stage with 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’.
That means the shows, which will take in venues in Australia, Norway, South Africa, and the UK, will not be able to use clips from the show or feature The Stig.
There are rumours the three men intend to team up to continue their careers together after Clarkson’s exit from the BBC, but sources close to the tour say it is just a matter of fulfilling commitments to the fans.