If memory serves me correct, Bufalino was wrongly identified as a Lucchese member by the FBI in the early 1960s. He had strong ties to and was closely associated with Jimmy Plumeri (a Lucchese member) and his connections to the Teamsters which could be the reason why the FBI wrongly ideed him with that Family. From the mid 1960s onwards however, realizing their blunder, Bufalino´s name was not shown on any FBI Lucchese member lists.
A book entitled Hoffa´s War was released around 1980. In the book, the author claims that Bufalino had served as an top advisor for the Genovese bosses. I don´t remember the details but I think the author stated that he had used FBI documents dated in the 1970s as basis for the Bufalino-Genovese-advisor claim. In my opinion, most likely, the FBI was wrong about Bufalino again. If anybody have more info on this, please correct me if I´m wrong.

My point is, and to answer your question, that the rumor of Bufalino somewhat being involved with the leadership of the Genovese Family did not start on the internet, but most likely stemms from Hoffa´s War, the book.