This week, BBC director general Mark Thompson defended Clarkson after the Top Gear host had prompted more than 32,000 complaints for saying on the BBC One show that striking workers should be "executed in front of their families".
Thompson told MPs on the Commons culture, media and sport select committee that Clarkson's comments were made "entirely in jest", and insisted that he would not be sacked over a "couple of flippant remarks".
But former Springwatch presenter Oddie said that the corporation's attempt to brush off the controversy is in stark contrast to its strong response to the 2008 'Sachsgate' scandal on BBC Radio 2, in which Russell Brand and Jonathan Ross left lewd messages on the answephone of actor Andrew Sachs.
The controversy led to Brand and Radio 2 controller Lesley Douglas resigning from the corporation, and Ross being suspended.
"I am tempted to say, 'What is the difference between that sort of public statement and what Jonathan Ross and Russell Brand did?'" Oddie told The Daily Telegraph.
"OK, that was more personal, but then it was one person… [Clarkson has] upset half the bloody country with this one."
Clarkson said that he was prepared to apologise for any offence caused by his comments, but he also insisted that they were merely intended as a joke.
Oddie, however, was unconvinced by that.
"For him to then say it was a joke is equally crass, because it isn't a joke and, if it is, where's the punchline, please? It's a reaction; he just can't help it. It's something that happens, but we don't need that," said Oddie.
"I don't think he should, necessarily, go, though. I didn't think the total outcome of the Ross and Brand thing was correct - people resigned and shouldn't have had to.
"Jeremy's a highly intelligent bloke, he's brilliant at his job, and has a lot going for him. I always think he's like a panto villain. Unfortunately, his brain doesn't engage at moments like that."
Clarkson has often poked fun at Oddie during motoring show Top Gear, taunting the former Goodie over his love of bird watching and ecology.
"I remember him saying silly things like, 'If Bill Oddie had his way, we won't have any cars, we'll all be walking everywhere and we'll grass over the car parks'," said Oddie.
"As it happens, I couldn't be less of a car freak. It's childish, actually, and it's name-calling.
"I have met him once or twice, briefly. At one of our meetings, he was at great pains to tell me how he's taken up birdwatching, and, not joking, what he had in his garden."