Green spent the past two seasons in Phoenix and played with Dragic, who was acquired by the Heat in February as part of a three-team trade.
"The vision is clear: Try to battle for the top spot in the NBA," Dragic told reporters at a news conference. "With this team, I think it's going to be unbelievable. I think a lot of opponents are going to be scared of us. Every player on this team is smart, and every player is really locked into one goal, and that's to win a championship."
Heat president Pat Riley also said he expects to contend in the Eastern Conference next season.
"With a complete roster, we can contend in the East, and I mean contend high. We're gonna go for it," he told reporters. "You have to have high expectations. We have moved on from what happened last year."
Green has averaged 10.2 points in 428 regular-season games with the Suns, Boston, Minnesota, Houston, New Jersey and Indiana, mostly coming off the bench.
He has made 341 shots from 3-point range over the past two seasons, the ninth most in the NBA during that span.
The Associated Press contributed to this report.