The suspensions will come as a result of the ongoing investigation into the Miami-based Biogenesis clinic, which allegedly supplied players with performance-enhancing drugs.
The details of the possible suspensions are not yet known, but it is possible 100-game bans might be on the table for Braun and Rodriguez.
Braun suffered injuries this season, appearing in only 58 games for Milwaukee. As of Tuesday, he was batting .304 with nine home runs and 36 RBI.
Rodriguez, who had recently undergone arthroscopic hip surgery in January, began a minor-league rehab assignment last week and has not played for the Yankees yet this season.