He later posted a video response to the allegations, in which he explains he did three takes of the scene, and in two of them he "mumbled" where the offensive word should have been.
However when he watched the footage back, he realised if "you listen very carefully, with the sound turned right up" it did sound like he had used the word he was trying to obscure.
"I was mortified by that. I was horrified. It is a word that I loathe," he says in the video.
He wrote to producers of the shows, explaining he did not use the word, although it may have sounded like he did, and he asked for another version of the scene to be used.
Last week, the BBC apologised over a remark made by Clarkson during another episode of Top Gear.
While building a makeshift bridge over the River Kwai in Thailand, Clarkson said: "That is a proud moment - but there's a slope on it."
A Top Gear executive producer said: "When we used the word 'slope' in the recent Top Gear Burma Special it was a light-hearted word play joke referencing both the build quality of the bridge and the local Asian man who was crossing it.
"We were not aware at the time, and it has subsequently been brought to our attention, that the word 'slope' is considered by some to be offensive and although it might not be widely recognised in the UK, we appreciate that it can be considered offensive to some here and overseas, for example in Australia and the USA.
"If we had known that at the time we would not have broadcast the word in this context and regret any offence caused."
A list of Jeremy's "best" moments
April 2014: Named his black terrier Didier Dogba after the Ivory Coast footballer.
March 2014: Joked a bridge with an Asian man walking across it had a 'slope on it'. His Top Gear producer Andy Wilman claimed Clarkson did not realise it is an offensive term.
January 2012: Made a series of derogatory remarks about India and ridiculed the sanitation for its poor by driving around slums in a Jaguar fitted with a toilet.
January 2012: Compared synchronised swimming with the deaths of 23 Chinese cockle pickers in Morecambe Bay in 2004.
July 2011: Said his new TV's energy-saving setting made it seem as if programmes were being 'presented by Lenny Henry in a cave'.
February 2011: Joked about Mexicans, prompting complaint by ambassador.
August 2010: Attacked by disability charities after he said the F430 Speciale car should be called 'Speciale needs'.
October 2009: Accused TV producers of being fixated with having 'black Muslim lesbians' on programmes.
February 2009: Described Gordon Brown as a 'one-eyed Scottish idiot'.
April 2007: Deriding a car made in Malaysia he suggested it was built by 'jungle people who wear leaves as shoes'.
December 2005: Gave a Nazi salute while presenting a piece about German firm BMW. Said its satnav 'only goes to Poland'.
- Independent, Daily Mail