Any changes would have to be agreed to by both the union and the league.
"That's an active discussion right now," Weiner said. "There are certainly some players who have expressed [a desire for tougher penalties], some of them after the Miami [Biogenesis] allegations.
"We are at a point, and it's clear where the majority of players are: They want a clean game, and we want to make it clear to players that is where the majority is at."
Weiner, who was at Blue Jays camp in Dunedin, Fla., as part of his Spring Training tour, pointed out that MLB already has the toughest penalties of any team sport. Tests for human growth hormone are now included.
"We focused this offseason on making it likely that more players are going to get caught," Weiner said. "I would explain that to players, what we've done with respect to HGH and testosterone testing. I'm not surprised there is greater frustration from players about the Miami stories, because, again, players are sick of this issue. It's natural for a lot of guys to say we need stiffer penalties.
"We've had a good discussion about whether that's fair to treat players who make a negligent mistake the same as players who intentionally try to beat the system, whether or not 50 games for a first time is sufficient, and we'll discuss that over the course of 2013."
The Players Association plans to talk to all players connected to Biogenesis.
"We'll speak with all the players whose names have surfaced," Weiner said. "And again, their names have only surfaced in newspaper articles."
Evan Drellich is a reporter for MLB.com. Follow him on Twitter @EvanDrellich. This story was not subject to the approval of Major League Baseball or its clubs.