The most oft-repeated story is that Puzo happened to be at a birthday party at Chasen's restaurant when Sinatra happened to be there. Puzo actually got up and introduced himself to Sinatra, and "all hell broke loose," with Sinatra supposedly threatening Puzo. A rumor had it that Sinatra actually applied for a part in the movie (not Johnny) but I doubt it.
Probaby the reason Sinatra didn't sue was that doing so would only call more attention to the similarities between him and Johnny Fontaine. But in the film, the similarities are greatly toned down compared with the details presented in the novel. Probably few people knew the rumor about Willie Moretti as the real-life model for Luca Brazi in intervening to help the singer's career. If Sinatra made a big stink, more people would have found out about it.