The BBC said the letters were sold separately but bought by one bidder, who wished to remain anonymous.
Bamfords said Wilde appears to be trying to romance magazine editor Alsager Vian in the letters at a time when homosexuality was illegal.
"The letters are very important as they help to fill in pieces of Oscar Wilde's tempestuous jigsaw," Alan Judd from Bamfords told the BBC.
Wilde wrote the plays "The Importance of Being Earnest" and "An Ideal Husband" and the novel "The Picture of Dorian Gray."
He died of cerebral meningitis in 1900 at the age of 46.