Clarkson, 54, apologised after a newspaper report earlier this week revealed he used an epithet for blacks while reciting the rhyme "Eeny, Meeny, Miny, Moe" to choose between cars while filming two years ago.
The presenter, known for his humorous but brash, blunt style, released a video on Twitter on Thursday apologising and saying he had tried to avoid the racist expression used in a well-known older version of the rhyme.
He said he recorded three takes, mumbling the word "nigger" in two versions and substituting it with "teacher" in a third, and only realised later that the word could be heard clearly in one take.
"I was mortified by this, horrified. It is a word that I loathe and I did everything in my power to make sure that version did not appear in the programme," said Clarkson, seen in the video brandishing a note to this effect sent to producers.
"As I am sitting here begging your forgiveness for the fact ... obviously my efforts weren't quite good enough."
The version did not appear on the show and it was not clear how the Mirror newspaper, which first published the story, got a copy of the take.
The revelation prompted calls for Clarkson to resign or be sacked by the BBC, which is publicly funded by an annual licence fee of 145.50 pounds ($250) that must be paid by all UK households with a television.
"I don't think the taxpayer should be subjected to people like Jeremy Clarkson who uses and insults people by his comments. It is time for him to go," opposition Labour MP Jim Sheridan from parliament's Culture, Media and Sports Committee said on Friday.
BBC SAYS VIEWS ISSUE SERIOUSLY
Top Gear is one of the BBC's best-selling shows globally, broadcast in 214 countries. The show have made Clarkson, a journalist who writes newspaper columns as well as a presenter, into a global celebrity.
His strong views have pitched him up against politicians, national governments, green groups, car companies and communities across Britain over the years. Facebook has an "I Hate Jeremy Clarkson" page.
But the BBC has regularly downplayed any controversies about his comments and trod carefully this week. A spokeswoman said in a statement issued late on Thursday the corporation had spoken to Clarkson.
"We have made it absolutely clear to him, the standards the BBC expects on air and off. We have left him in no doubt about how seriously we view this," she said.
She declined to comment further on Friday and a spokesman for Clarkson told Reuters that the presenter was not expected to make any further comment.
The annual report of BBC Worldwide - the broadcaster's overseas sales branch - says Top Gear notched up the corporation's third title in the Guinness World Records, where it was named the world's most widely watched factual TV programme in 2013.
The show, which started in 1988, has sparked various spin-offs including video games, a game app and Top Gear magazine.
During 2012/13, the BBC said it had agreed new commercial deals with UK presenters Clarkson, James May and Richard Hammond to ensure the show's future for the next three years.
The BBC declined to say how much Clarkson makes or how much "Top Gear" and its spin-offs earn for the company.