Marc Rimbaud is the current artistic director of the Royal Swedish Ballet.
He has also worked as the director of the Royal Winnipeg Ballet.
Ashton was appointed Artistic director of the Royal Ballet from 1963 to 1970, when he retired from the post.
He was artistic director of the Royal Ballet in London between 1970 and 1977.
"I am very disappointed that the tour will not take place," Anthony Dowell, the director of the Royal Ballet, said in a statement.
He was appointed as artistic director of the Royal Ballet in 1986, and held the position until 2001 when he officially retired.
His first term as artistic director of the Royal Danish Ballet was from 1895-1994.
Four years later the young Bournonville, who had become the director of the Royal Danish Ballet in 1829, created his own version.
In 1986 Sir Anthony became the fifth director of the Royal Ballet, founded in 1931.
He was appointed director of the Royal Winnipeg Ballet in 1957.