Braun refused to answer questions at a meeting with MLB about his connection to Tony Bosch and the Biogenesis of America, the Miami clinic at the center of the banned substance scandal that came to light in February.
Rodriguez will reportedly have a similar meeting Friday, and is also expected to refuse to answer questions. A source said another 10 players linked to Bosch and Biogenesis have already met with the league.
The source said MLB Commissioner Bud Selig planned to hand down suspensions to Braun, Rodriguez and the other players as soon as next week, after the regular season resumes Friday following the All-Star Game.
Braun and Rodriguez may receive 100-game suspensions, the punishment for second offenses, based on the accusation that both players had committed several offenses by accepting the banned substances from Bosch and then lying about it.
Bosch agreed to cooperate with MLB, turning over documents, phone, text and email records.
Braun was awarded the National League MVP in 2011, even after his testosterone tested "insanely high, the highest ever for anyone who has ever taken a test, twice the level of the highest test ever taken." He was able to successfully challenge the test, and its accompanying 50-game suspension, based on "a technicality" on how the sample was handled.
Rodriguez was involved in a 2003 report involving more than a hundred players testing positive to using anabolic steroids and other banned drugs that was conducted anonymously. He received no punishment at the time, and has admitted to using drugs between 2001 and 2003.
Other players reportedly involved in the case are Everth Cabrera, Fancisco Cervelli, Nelson Cruz, Melky Cabrera, Bartolo Colon, Yasmani Grandal, Fernando Martinez, Jesus Montero, Jordan Norberto, Jhonny Peralta, César Puello and Fautino de Santos.