Greenberg, whose group includes Rangers President and retired all-star pitcher Nolan Ryan, has 30 days during which only he can negotiate a purchase.
Terms of the deal, announced late Tuesday, were not disclosed but analysts had expected the club to attract bids of at least $500 million to $550 million.
Under the deal, Hicks, who is selling the team to satisfy creditors who declared his sports group in default, would retain a "significant" interest in the Rangers. Hicks Sports Group (HSG) also owns the Dallas Stars hockey team.
"There's a lot of work to do," Greenberg said in a statement. "The deal isn't done yet, but I am confident we can complete this soon."
Major League Baseball, which has loaned Hicks money to operate the team, had set a Tuesday deadline for one of the three groups bidding for the team to be picked by Hicks for exclusive negotiations on a purchase.
Greenberg's victory is not complete yet as he must finalize terms of a deal, and garner the approval of 75 percent of Major League Baseball's 30 owners as well as the HSG creditors.
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, and sold another team in Altoona, Pennsylvania, last December. He also owns Greenberg Sports Group, a sports consulting firm.
Greenberg beat out groups led by Houston businessman Jim Crane and former sports agent Dennis Gilbert.
Crane, the former CEO of freight-forwarding company EGL Inc, re-entered that industry in August 2008 with the formation of Crane Worldwide Logistics. He also bid for the Chicago Cubs, which bankrupt media company Tribune Co sold this year for $845 million.
Gilbert is a special assistant to the owner of the Chicago White Sox baseball team. He is a former baseball agent whose firm once represented baseball home run king Barry Bonds.
In April, 40 creditors - banks and institutional investors - declared HSG in default on $525 million of loans after Hicks withheld a quarterly interest payment.
Hicks also owns half of the English Premier League's Liverpool Football Club, which is held separately from HSG.
