Griffin said Beltre first injured his thumb last season, and "has played this whole year with a torn ligament."
Surgery apparently will be needed to repair the damage, and it could happen sooner rather than later.
"We just want to make sure that if anything has to take place, that there's plenty of recovery time," manager Jim Riggleman said. "So we're getting a little head start on it. Like with [Jeff] Clement and get it done sooner than later."
Clement, who had been bothered for more than a month with a sore left knee, underwent arthroscopic surgery Tuesday to have his lateral and medial meniscus repaired. He is expected to be 100 percent in plenty of time for Spring Training.
"He's resting comfortably at home," Griffin said.
Beltre, who has played through various ailments this season, is batting .268 with 25 home runs and 74 RBIs. He became the fourth player in franchise history to hit for the cycle Sept. 1 against the Rangers.
His offensive production has been affected by a sore right shoulder and leg discomfort.
"I've been with [Beltre] in the Dodger organization for four years and up here, and it's the same thing," Riggleman said. "He didn't miss games down there, and he doesn't miss games here.
"He has a high pain threshold. When the trainers and doctors check him out, they say, 'Man, we can't believe this guy is playing.' He takes a lot of pride in the fact that he signed a contract to play the season, and he does everything he can to do it."
Left-handed pitcher Erik Bedard, who has been out of action since July 4 with a shoulder ailment, threw 30 pitches off the bullpen mound with the catcher crouching and came through the session well, according to Griffin.
"He threw on a downward plane at about 70 percent and said he felt good," the trainer said. "He will throw another bullpen in three days, and another one three days after that with increased intensity and more pitches and we'll see how he feels. Instead of day-to-day, it's bullpen-to-bullpen."
Left-hander Jarrod Washburn, sidelined with an abdominal strain, played catch without pain for the first time prior to Tuesday night's game.
"We will proceed cautiously, make a decision on when he will throw some bullpens and go from there," Griffin said.