"It's a complex business deal that positions the franchise positively for the future," Tom Hicks, whose family controls Hicks Sports Group (HSG), said in a statement.
Because the transaction is still subject to Major League Baseball approval and other matters, specific terms were not announced.
At least 75 percent of Major League Baseball's 30 owners as well as HSG creditors must approve the sale.
Analysts have expected the club to bring $500 million to $550 million.
In a separate transaction, the Hicks-controlled Ballpark Real Estate has agreed to sell or transfer roughly 153 of the 195 acres it owns surrounding the Rangers and Cowboys stadiums to the Greenberg-Ryan group.
Ballpark Real Estate will receive cash, notes, and an ownership position in the team, the Rangers said.
Hicks, who also owns the NHL's Dallas Stars and half of the English Premier League soccer club Liverpool, is selling the Rangers to satisfy creditors who in April declared his sports group in default on $525 million of loans after the Texan withheld a quarterly interest payment.
Greenberg helped negotiate hockey star Mario Lemieux's purchase of the Pittsburgh Penguins hockey team in 1999. He owns minor league baseball teams in Myrtle Beach, South Carolina, and State College, Pennsylvania.
Greenberg will serve as managing partner and CEO of Rangers Baseball Express while Ryan will continue as president.
(Reporting by Gene Cherry in Salvo, North Carolina; Editing by John O'Brien)