The episode, which was filmed in Burma and Thailand and shown in March, featured a scene where the motoring show's stars 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 phrase was an example of "casual racism" and "gross misconduct".
The BBC Two 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 BBC One'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.