They earned the final three spots that opened after a rash of injuries knocked out at least four players who would have been on the team.
They beat out Eric Gordon, Rudy Gay and No. 1 NBA draft pick Anthony Davis of New Orleans, who couldn't practice this week because of a sprained ankle.
The person spoke on the condition of anonymity because the official team announcement was not to come until late Saturday.
Also heading to London for the defending gold medalists are: LeBron James, Kobe Bryant, Carmelo Anthony, Chris Paul, Deron Williams, Kevin Durant, Russell Westbrook, Tyson Chandler and Kevin Love.
Griffin showed he was healthy again after being slowed by a knee injury during the playoffs, putting on an impressive dunking display following practice on Saturday.
Harden, the NBA's Sixth Man of the Year with Oklahoma City, gives the Americans more scoring punch off the bench, and Iguodala is a defensive specialist who can guard multiple positions.
The Americans lost Dwight Howard, Derrick Rose, Dwyane Wade and Chris Bosh to injuries in recent months, forcing them to scrap plans to name their 12-man team on June 18.
They asked the U.S. Olympic Committee for an extension and added Harden and Davis to their original pool of finalists that was chosen in January.
USA Basketball chairman Jerry Colangelo was intrigued by Davis' skills, but the college player of the year at University of Kentucky sprained his ankle last week and was not able to take part this week, robbing him of a chance to show he was ready to take part in international competition now.
James, Bryant, Anthony, Paul and Williams were all part of the 2008 Olympic gold medalists. Durant, Westbrook, Chandler and Love played on the world championship team two years ago.