Jones, a Miami native, was signed as a free agent this summer to give the Heat a 3-point shooter the team lacked last season, when it stumbled to the NBA's worst record. He'd been slowed throughout camp because of a sprained wrist, but even this week, the team was hopeful his return would come in a matter of days.
Jones ruptured the tendon while shooting on Wednesday.
"Just imagine somebody punching you in your stomach as hard as they can," Jones said. "I felt like I was turning the corner."
The surgery will be performed Saturday.
Jones averaged eight points last season for Portland, and his 44 percent shooting from 3-point range ranked second in the NBA.
Jones is the second Miami player to be lost for an extended period because of injury so far this season. Center Jamaal Magloire will miss six to eight weeks because of a broken left hand that will not require surgery.
The Heat may try to make a move to add a 3-point shooter to the roster before the preseason ends.