Agent Mark Bartelstein says Rush will hold the option in the second season of the deal.
Rush was Golden State's top reserve during the 2011-12 season before tearing the anterior cruciate ligament in his left knee in the team's 2012-13 home opener. He was traded to Utah last summer as part of a salary-shedding move that allowed the Warriors to sign Andre Iguodala.
Rush played just 38 games for the Jazz last season after returning from that devastating knee injury. He held a workout for teams in Las Vegas recently before signing with the Warriors, who had pursued Rush from the start of free agency July 1.
"I think he showed everybody his game is coming back," Bartelstein told the Salt Lake Tribune. "He's gotten more and more explosive week by week."
Rush has averaged 8.3 and 3.3 rebounds in six seasons.
The Associated Press contributed to this report.