Clarkson became embroiled in a racism row following claims that he used the n-word while reciting the nursery rhyme Eeny, Meeny, Miny Moe during shooting for an episode of the BBC2 programme.
He said: "I wish to God that my attempts to cover up that word were better than they were.
"I was simply mumbling - saying 'ner ner' or something similar, anything but the N-word. It was my mistake and I apologise for not covering it up. But if you look at the footage you can see what I'm trying to do."
In the footage Clarkson is using the nursery rhyme to compare two sports cars. He said he "mumbled where the offensive word would normally occur" in two takes, and used the word "teacher" in its place in a third.
The segment was later edited out of the BBC broadcast.
He said he was unable to think of another way to compare the cars.
"I've racked my brains to think if there is another way to do it without doing eenie, meenie, miny, mo. It was a good hour.
"What we did was entirely innocent. I would invite my readers to tell me of an alternative."
Clarkson also took to Twitter last night to tell fans he is "begging for their forgiveness".
In the video post a solemn-looking Clarkson said: " I was mortified by this, horrified. It is a word I loathe.
"Please be assured I did everything in my power to not use that word. And as I'm sitting here begging your forgiveness for the fact that obviously my efforts weren't quite good enough."
He added: "When I viewed this footage, several weeks later, I realised that if you listen very carefully with the sound turned right up, it did appear I actually used the word I was trying to obscure," he added of one of the first two takes.
He said the item was recorded a "couple of years ago" and he "did everything in my power to make sure that that version did not appear in the programme that was transmitted".
Clarkson added that he sent a note to the production office at the time, asking if there was another take that could be used.
"Jeremy Clarkson has set out the background to this regrettable episode," the BBC said.
"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."
The allegations were reported in Thursday's Daily Mirror, which claimed the footage was studied by audio forensic experts who told them the star can be heard mumbling the n-word.
Clarkson had earlier tweeted: "I did not use the n-word. Never use it. The Mirror has gone way too far this time."
His co-host, James May, also came to his defence on Twitter, saying: "Jeremy Clarkson is not a racist. He is a monumental bellend and many other things, but not a racist. I wouldn't work with one."
The claims come days after the motoring show's producer apologised for broadcasting a "light-hearted" joke by Clarkson that sparked a complaint of racism.
An episode, filmed in Burma and Thailand and shown in March, featured a scene in which the presenters built a bridge over the River Kwai, and as an Asian man walked over it Clarkson said: "That is a proud moment, but there's a slope on it."
Somi Guha, an actress who complained to the BBC, said the use of the word "slope" was an example of "casual racism" and "gross misconduct".
The BBC2 show's executive producer Andy Wilman 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."
Clarkson is well known for courting controversy - in recent years he has been cleared of breaching the broadcasting code by watchdog Ofcom after comparing a Japanese car to people with growths on their faces.
He previously faced a storm of protest from mental health charities after he branded people who throw themselves under trains as ''selfish'' and was forced to apologise for telling BBC1's The One Show that striking workers should be shot.
The motoring show has also faced complaints from Indian and Mexican politicians over remarks made about their countries while filming on location.