Well, as Gravano said, they didn't feel good going to the Genovese family because Paul and Chin were very tight. Obviously they didn't bother to approach the other actual bosses either - Anthony Corallo, Carmine Persico, or Rusty Rastelli, who, like the Chin, would have also likely been against the hit on their fellow boss. But those guys were going away because of the Commission case and other convictions so apparently Gotti felt OK going to the younger up and comers in those families - Vic Amuso and Anthony Casso, Gerry Langella, and Joe Massino. But of course Chin was still very much in control of the Genovese family and going to them with permission would have probably exposed the whole plot.

And they ended up being right. Chin was pissed that Paul had been killed without Commission approval. That kind of hostile takeover was a threat to other bosses, including himself. When the Gambinos told the other families they had nominated Gotti to be the next boss, the Genovese sent word back that they would accept it but that "someday somebody would have to pay for Paul's murder."

Of course that didn't take too long, with Gotti's underboss Frank DeCicco (who had been in on the plot) being killed in the car bomb a few months later; the bomb was primarily designed for Gotti. Then a couple of years later Genovese consigliere Bobby Manna and some others were overheard on an FBI bug plotting to kill both John and Gene Gotti. The FBI paid a visit to Gotti's house to warn him of the plot.

Agent Bruce Mouw said they were there on official business, that they had information that there was a plot on his life. Gotti replied that he didn't have any problems with anyone but his expression changed when Mouw said it was the Genovese family. "He blinked when I said the West Side," Mouw later recalled. "That meant something to him. This guy fears Chin."