Citing unidentified sources, the CBS news show said the leak occurred in February, days after the Miami New Times published documents implicating the Yankees star in the Biogenesis investigation.
In the Miami New Times documents, the names of Braun and one of Rodriguez's teammates, catcher Francisco Cervelli, are redacted. "60 Minutes" reports that members of Rodriguez's camp obtained unredacted versions and leaked them to Yahoo! Sports.
Rodriguez's lawyer, David Cornwell, denied the allegations to "60 Minutes."
"The allegations are untrue and are another attempt to harm Alex — this time by driving a wedge between Alex and other players in the game," he said in a statement to the show. "While Alex focuses on baseball and repeatedly states that he is going to respect the appeal process, the drumbeat of false allegations continues."
Rodriguez has been suspended for 211 games through the 2014 season. He is allowed to play until a decision is issued by arbitrator Fredric Horowitz, which is not expected until at least November. Braun, the Milwaukee Brewers slugger, accepted a 65-game suspension last month.
All-Stars Nelson Cruz, Jhonny Peralta and Everth Cabrera were banned 50 games apiece Aug. 5 when Major League Baseball disciplined 13 players, including Cervelli, for their relationship to Biogenesis of America, a Florida anti-aging clinic accused of distributing banned performance-enhancing drugs.
Rodriguez admitted four years ago that he used PEDs while with Texas from 2001-03 but has repeatedly denied using them since. His penalty was more than double the previous high for a PED suspension.
A-Rod's drug penalty was for "his use and possession of numerous forms of prohibited performance-enhancing substances, including testosterone and human growth hormone over the course of multiple years," MLB said.
His punishment under the labour contract was for "attempting to cover up his violations of the program by engaging in a course of conduct intended to obstruct and frustrate the office of the commissioner's investigation."
Commissioner Bud Selig said Thursday he thought the punishment was "eminently fair."
"I have a job to do, and that's protecting the integrity of the game and enforcing it, and that's what I'm going to do," he said.
MLB's investigation began last year after San Francisco outfielder and All-Star game MVP Melky Cabrera tested positive for elevated testosterone, as did Oakland pitcher Bartolo Colon and San Diego catcher Yasmani Grandal. The inquiry escalated in January when the Miami New Times published documents obtained from former Biogenesis associate Porter Fisher that linked several players to the clinic.
In June, baseball struck a deal for Biogenesis founder Anthony Bosch to co-operate. After holding investigatory interviews with the players, MLB presented evidence to the players' union along with its intended penalties, starting the final round of negotiations.