Top Gear presenter Jeremy Clarkson says the BBC has told him he will be sacked if he makes "one more offensive remark, anywhere, at any time".
Although he mumbles the word, Clarkson begins by saying the letter "n".
The BBC says it "left him in no doubt about how seriously we view this".
Clarkson was initially accused of using the "n-word" on Thursday by the Daily Mirror, which said it had hired "audio forensic experts" to analyse the clip it had obtained.
The presenter initially told his 3.3 million Twitter followers: "I did not use the N-word. Never use it. The Mirror has gone way too far this time."
Later that day after the newspaper posted a clip of the incident - which was filmed in 2012 and never broadcast - he released a video statement "begging forgiveness" for the error.
This time, he admitted he had appeared to "mumble" the offensive word despite attempting not to.
Clarkson wrote in the Sun, where he has a weekly column: "I've been told by the BBC that if I make one more offensive remark, anywhere, at any time, I will be sacked.
"And even the angel Gabriel would struggle to survive with that hanging over his head.
"It's inevitable that one day, someone, somewhere will say that I've offended them, and that will be that."
He also said the BBC had told him "very firmly" to apologise but added: "Apologising for using the n-word would be the same as apologising for starting the war in Syria. It's something I hadn't done."
Clarkson added: "I use the F-word pretty much constantly and the C-word too, especially when I'm talking about James May. But the N-word? No. It's not in my lexicon."
He also highlighted that the expert used by the Daily Mirror had told LBC that she could only be 75% certain the word was used.
Michelle Bowman of digital forensics company CY4OR told Nick Ferrari: "You can't be 100% certain, it's not an exact science. Ideally you would want to compare that phrase with a phrase where the word is said or where a different word is said."
Although the clip was never broadcast on the BBC Two show, the corporation said it had received more than 300 complaints following recent media coverage. On Saturday, the BBC said it had nothing to add to its earlier statement.
On Thursday it said: "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."
Deputy Labour leader Harriet Harman has called for the BBC to sack the presenter.
Writing on Twitter Ms Harman said: "Anybody who uses the N-word in public or private in whatever context has no place in the British Broadcasting Corporation."
The prime minister - a friend of Clarkson's - felt it was "absolutely right that there has been an apology", his spokesman said.
But he refused to comment on whether Clarkson should lose his job, saying: "His view is that in terms of actions and the like, that's for the BBC".
It is not the first time the Top Gear presenter has been accused of racism.
The BBC apologised in 2011, after an episode in which co-presenter Richard Hammond called Mexicans "feckless [and] flatulent" and Clarkson joked they would not receive complaints because the Mexican ambassador would be asleep.
Apologies were also made for an episode broadcast in March, in which Clarkson used the word "slope" as an Asian man crossed a newly built bridge over the River Kwai in Thailand.
The use of the word - which is a derogatory term for people of Asian descent - led to complaints.