The episode, filmed in Burma and Thailand and shown in March, featured a scene where the 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 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 the build quality of the bridge and the local Asian man crossing it.
"It has subsequently been brought to our attention that the word slope is considered by some to be offensive. If we had known at the time we would not have broadcast the word in this context and regret any offence caused."