Professional ballroom dancer Anton Du Beke called television actress Laila Rouass, who has a Moroccan father and Indian mother, by the insulting term a fortnight ago, The News of the World tabloid said.
The two are partners in the celebrated BBC television programme "Strictly Come Dancing" and Du Beke used the term after the former star of "Footballers' Wives" had a spray tan, the paper said.
The News of the World quoted an eyewitness as saying Du Beke, 43, told Rouass: "Oh my god, you look like a Paki" in the presence of 15 other people, who were left "gobsmacked".
"Then Laila glared at Anton and just walked out - she was so upset. She just went to her dressing room, collected her stuff and went straight home. She felt totally humiliated, " the witness said.
Offering an unconditional apology, Du Beke said: "I must say immediately and categorically that I am not a racist and that I do not use racist language.
"During the course of rehearsals Laila and I have exchanged a great deal of banter entirely in jest, and two weeks ago there was an occasion when this term was used between the two of us.
"There was no racist intent whatsoever but I accept that it is a term which causes offence and I regret my use of it, which was done without thought or consideration of how others would react. I apologise unreservedly for any offence my actions might have caused."
Rouass accepted Du Beke's apology, saying: "It was a situation which happened that we have moved on from and I accept his apology. I'm really enjoying the show and dancing with Anton and hope we can go as far as possible in the competition."
A BBC spokesman said: "The BBC does not condone offensive language in the workplace."
In February, former prime minister Margaret Thatcher's daughter Carol, a journalist, was forced to quit her job at the BBC after referring to a French-Congolese tennis player as a "golliwog" in the green room.