Jeremy Clarkson has begged viewers' forgiveness after he appeared to use the N-word during filming of his BBC programme Top Gear.

In a video statement posted online on Thursday, he said that he had tried to obscure the word when reciting the "eeny, meeny, miny, moe" nursery rhyme to chose between two cars, but that his efforts to do so "weren't quite good enough".

Clarkson had previously issued a robust denial of the allegation, telling his 3.2 million Twitter followers: "I did not use the N-word. Never use it. The Mirror has gone way too far this time."

But footage of the incident was later posted on The Daily Mirror website. In it Clarkson recites the beginning of the children's nursery rhyme "eeny, meeny, miny, moe" before apparently mumbling: "Catch a nigger by his toe."

His apology came after a day of growing calls for the BBC to sack him. The corporation issued a strongly worded statement saying: "Jeremy Clarkson has set out the background to this regrettable episode. 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.".

Downing Street condemned any use of the word, saying that David Cameron – a friend of Clarkson's – would "certainly not" use it.

Aliya Mohammed, the chief executive of Race Equality First, called for immediate action from the BBC.

She said: "I am appalled at yet another open display of racism from Clarkson. How many racist comments will the BBC allow from the presenter? It makes me question whether Clarkson feels it would be acceptable because he thinks viewers and others within the BBC share his views.

"Racism is illegal and it has no place in modern society. I urge as many people as possible to show their condemnation of this abhorrent display of racism. A public personality should set an example to viewers and Clarkson is clearly a very bad example."

Lawyers for the Indian-born actor Somi Guha, who appeared in The Bill and other television shows, made a formal complaint to the BBC Trust as calls for Clarkson to be sacked gathered pace. In the letter, which was also sent to the media regulator Ofcom and has been seen by the Guardian, the lawyer Lawrence Davies demanded an external investigation into how the offensive language had come to be edited out of the show and whether Clarkson had been reprimanded.

Davies, director of the law firm Equal Justice Solicitors, said Clarkson would be dismissed in any other line of work. "It's appalling. He's a repeat offender and should be sacked," he said.

"If he was a Ukip councillor in a private meeting I believe they would suspend or dismiss him. If he was the owner of an American basketball team he would be fined and banned. But the BBC consider themselves unaccountable on the issue of racism."

The prime minister's spokesman said: "In terms of any usage of that word, that would be quite wrong."

He added that the BBC would be investigating the claims. "If there are those types of reports and allegations I'm sure that is something the BBC will look at."

Clarkson, who earned more than £14m last year for his Top Gear work, is understood to have spoken to senior BBC figures on Thursday morning as part of the corporation's investigation into the allegations.

Executives learned about the controversy only when a preview of the Mirror's front page was published online on Wednesday night.

Clarkson's Top Gear co-host James May earlier leapt to the star's defence. May told his 1.4 million Twitter followers: "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. #ThatIsAll."

Piers Morgan, the former CNN presenter with a long history of feuding with Clarkson, led calls on Twitter for him to be dismissed. He said Clarkson should be "TV toast", and added: "Ron Atkinson was fired by @ITV for using the N-word. Don't see how the BBC can avoid firing @JeremyClarkson if he did the same … In America, he'd be fired by now."

The programme, aired in February 2013, showed Clarkson standing between a Toyota GT86 and a Subaru BRZ and saying: "Both cost the same, they have the same bodies and the same interiors with the same equipment.

"They are even built by the same people in the same factory. You might imagine then that it's quite hard to choose between them, but actually, it isn't. Watch, because this is how you do it."

In the unaired footage – which was later passed to the Mirror – the presenter then recites the children's counting rhyme and appears to use the N-word under his breath before pointing at the Toyota and shrugging: "Toyota it is." The offensive term was removed from the episode, according to the Mirror. In the broadcast version, Clarkson recited the rhyme but this time used the word "teacher".

Clarkson's full apology

"Ordinarily I don't respond to newspaper allegations but on this occasion I feel I must make an exception. A couple of years ago I recorded an item for Top Gear in which I quote the rhyme "eeny, meeny, miny, moe". Of course, I was well aware that in the best-known version of this rhyme there is a racist expression that I was extremely keen to avoid. The full rushes show that I did three takes. In two, I mumbled where the offensive word would normally occur and in the third I replaced it altogether with the word teacher. Now when I viewed this footage several weeks later I realised that in one of the mumbled versions if you listen very carefully with the sound turned right up it did appear that I'd actually used the word I was trying to obscure. I was mortified by this, horrified. It is a word I loathe and I did everything in my power to make sure that that version did not appear in the programme that was transmitted.

"I have here the note that was sent at the time to the production office and it says: 'I didn't use the N-word here but I've just listened through my headphones and it sounds like I did. Is there another take that we could use?'

"Please be assured I did everything in my power to not use that word, as I'm sitting here begging your forgiveness for the fact my efforts obviously weren't quite good enough, thank you."