The Top Gear presenter was embroiled in fresh controversy earlier this week, after the Daily Mirror published unaired footage of Clarkson allegedly using the racist term 'n*****r' while reciting a nursery rhyme.
Clarkson issued a filmed apology via his Twitter page on Thursday.
In it, he admitted to having used the word, but said he did "everything in my power" not to.
"I'm sitting here begging your forgiveness for the fact my efforts obviously weren't quite good enough," he said in the film.
It comes after deputy Labour leader Harriet Harman urged the BBC to sack the presenter.
"Anybody who uses the N-word in public or private in whatever context has no place in the British Broadcasting Corporation," she said on Twitter.
But Education Secretary Michael Gove urged the corporation not to axe Clarkson because he had been "clear in his apology". And a spokesman for the Prime Minister added, "it is absolutely right that there has been an apology."
Clarkson is thought to have earned £14 million for his work on Top Gear last year, including his salary and payments related to the show's global success.
The presenter has been accused of racism before - only last week the BBC was forced to issue an apology after he appeared to have referred to an Asian man as a "slope" on a recent Top Gear episode.
Jeremy Clarkson's Top 20 Most Obnoxious Lines
"Short people. When you've finished using a car, put the f**king seat back, so humans can use it afterwards," he posted on Twitter. Offensively.
"The only person who looked good in a 4-seated convertible was Adolf Hitler."
On the Renault Clio V6 handling bends: "In typical French fashion it just gives up! A bit like the French did with the Germans."
"I'm thinking. If you had gone to the trouble of making a chemical bomb, why would you detonate it on a coach from Preston?"
On public sector workers in 2011: "I'd have them all shot. I would take them outside and execute them in front of their families."
"I don't understand bus lanes. Why do poor people have to get to places quicker than I do?"
"The only reason the Arabs and Jews have managed to keep their nasty little war going for 50 years is because it never bloody rains there. If the post-war powers had put Israel in Manchester, there'd have been no bloodshed."
"If all the creatures on earth were the same size, it's said a lobster would have the smallest brain. But then someone invented Wayne Rooney."
On the invention of the segway: "They're made in America, of course, so fat Yanks can go to the fridge without expending any energy."
"Supercars are supposed to run over Arthur Scargill and then run over him again for good measure. They are designed to melt ice caps, kill the poor, poison the water table, destroy the ozone layer, decimate indigenous wildlife, recapture the Falkland Islands and turn the entire third world into a huge uninhabitable desert, all that before they nicked all the oil in the world."
"I'm sorry, but having an Aston Martin DB9 on the drive and not driving it is a bit like having Keira Knightley in your bed and sleeping on the couch. If you've got even half a scrotum it's not going to happen."
On a caravaning holiday: "You aren't allowed to have a party, you aren't allowed to have music, you aren't allowed to play ball games, you aren't allowed to have a camp fire, you have to park within two feet of a post, you have to keep quiet, you have to be in bed by eleven. This is not a holiday, it's a concentration camp!"
"Britain's nuclear submarines have been deemed unsafe... probably because they don't have wheel-chair access."
"If we are being honest HIV is a pathetic virus, it can only live in the air for 6 seconds and it does what ebola does to you in 10 days in 10 years."
On the Lotus Elise: "This car is more fun than the entire french air force crashing into a firework factory."
On actress Sarah Jessica Parker: "People think 'oh she must be pretty'. She isn't - she looks like a boiled horse."
On the BMW Z3: "And if you are clinically insane, by which I mean you wake up in the morning and think you are an onion, here's your car."
On the Ferrari 430 Scuderia: "It's like God having really unusual sex... it should come with toilet roll."
"The problem is that television executives have got it into their heads that if one presenter on a show is a blonde-haired, blue-eyed heterosexual boy, the other must be a black Muslim lesbian."
"If you're thinking of coming to America, this is what it's like: you've got your Comfort Inn, you've got your Best Western, and you've got your Red Lobster where you eat. Everybody's very fat, everybody's very stupid and everybody's very rude - it's not a holiday programme, it's the truth."
-The Independent
