What, according to chapter 6 of The Great Gatsby, is the major rumor about Jay Gatsby's past and present associations?

There isn't one particular rumor that stands out regarding Jay Gatsby. There are actually a number of them, the most persistent being that he was living in a houseboat and secretly traveling up and down the Long Island shore. Most of the gossip revolved around his notoriety—that he was, in some way or another, connected to the criminal underworld. Another rumor which was also making the rounds was that he was attached to what was dubbed an "underground line to Canada"—probably some or other smuggling operation.

These stories obviously interested the tabloids, and one reporter, keen to make a name for himself, responded to one such rumor and approached Gatsby, who dismissed him without providing any detail. Jay, however, did not seem much bothered by the rumors and apparently actually enjoyed them.

When Tom and Daisy attended one of Jay's parties, Tom hinted at what he believed was Jay's involvement with crime:

“Who is this Gatsby anyhow?” demanded Tom suddenly. “Some big bootlegger?”

“Where’d you hear that?” I inquired.

“I didn’t hear it. I imagined it. A lot of these newly rich people are just big bootleggers, you know.”

Nick came to Jay's defense and Daisy later did the same after Tom suggested that he was going to investigate Jay's activities. She mentioned that he had made his money by starting a chain of drug stores. We learn throughout the novel that Jay definitely had criminal connections, the most obvious being Meyer Wolfsheim, who had, according to Jay, fixed the 1919 World Series.

When Nick later confronts Wolfsheim after Jay's death, he takes much pride in acknowledging that he had been Jay's mentor and had started him in business.

“I raised him up out of nothing, right out of the gutter. I saw right away he was a fine-appearing, gentlemanly young man, and when he told me he was at Oggsford I knew I could use him good. I got him to join up in the American Legion and he used to stand high there. Right off he did some work for a client of mine up to Albany. We were so thick like that in everything.'—he held up two bulbous fingers—'always together.'”

It is, therefore, ironic that Wolfsheim does not want to attend Jay's funeral since he had been the one responsible, in part, for his success. He disassociated himself from Jay, probably fearing that he would be scrutinized if their relationship were discovered.

Further irony lies in the fact that although Jay Gatsby had built up such notoriety, his death is not the result of a criminal act on his part, but of another. He is murdered by the vengeful Mr. Wilson, while the real perpetrator, Daisy Buchanan, escapes without even a blemish to her name.