Bennett was acquired from the Seattle Seahawks along with a seventh-round draft pick for a fifth-round selection and wide receiver Marcus Johnson in a deal that was announced shortly after the beginning of the new league year.
While the trade needed time to become official, Bennett didn't wait long to put others on notice.
"I know Eli Manning (of the New York Giants) is probably watching this and thinking ... yes, I'm coming. I know Dak (Prescott, of the Dallas Cowboys) is watching this like, 'Yeah, he's coming.' Yeah, I am," Bennett said on Monday. "And Alex Smith (of the Washington Redskins) he knows he can't run from me. I told him in the Pro Bowl.
"It's definitely going to be a great season, and it's going to be fun to be out here and be able to chase quarterbacks."
Bennett also praised his new teammates, telling reporters that he thinks the defensive line could be as dominant as a certain NBA team.
"I think it can be one of the greatest. I think we can have one of the greatest defensive lines to ever play the game if we approach the game like every single way, just go out there and keep doing what they're doing and just finding a way to just add and keep showing how many great players (we have)," he said. "And I think a great defensive line is about the rotation. It's kind of like Golden State, you want to be able to have those guys that can come in and shoot and shoot and score every time."
Bennett was owed a $3 million roster bonus had he been with the Seahawks on March 18. By trading him, the Seahawks saved $2.2 million against the salary cap.
The 32-year-old Bennett has battled nagging injuries, including foot and knee issues, despite making the Pro Bowl the past three seasons.
Bennett had 54 career sacks, including 8.5 sacks and 24 quarterback hits last season.