The program, The South Bank Show, will be broadcast by ITV.
Avril Bergen says that in the 1960s she heard overheard Fonteyn and Nureyev talking in a whisper about her pregnancy and miscarriage.
Although Nureyev was gay, he was believed to have had several sexual relationships with women. One of the program's producers told the Telegraph that "lots of people have gone on record about it for the first time."
Fonteyn and Nureyev were an intense and passionate performing pair, and there has long been speculation that such intensity arose from a romantic relationship, despite Nureyev's homosexuality, and despite the fact that Fonteyn was 19 years older.
According to Nureyev's personal assistant, Joan Thring, he wanted to marry Fonteyn.
Meredith Daneman, who published a biography of Fonteyn last fall, says she is not convinced by the new information.