Boston pitcher Daisuke Matsuzaka, center, walks off the field with a trainer, left, and his interpreter Tuesday in Seattle. Matsuzaka suffered a shoulder injury and was placed on the DL after the game, but hopes to be activated by June 12.

Baltimore ? Red Sox right-hander Daisuke Matsuzaka will begin a program to strengthen his sore right shoulder and hopes to be activated from the 15-day disabled list on June 12, the first day he is eligible to return.

“Just to have the exams done and to know for sure that everything’s fine is really good,” Matsuzaka said through a translator Saturday. “I believe I’ll be ready.”

Matsuzaka was placed on the DL on Friday after an MRI exam in Boston revealed a mild strain of his right rotator cuff. The move was retroactive to Wednesday. Matsuzaka is 8-0 with a 2.53 ERA in 11 starts this season.

“Even though I might feel OK, the club has told me to take this time to rest up and get healed well. Although I appreciate the rest and protection I’m getting, I do feel bad that I’m the only one getting rest at this time,” Matsuzaka said. “But dwelling on that isn’t going to help me heal any faster. It’s too bad that how hard I work out right now won’t affect the outcome of the game.”

Red Sox manager Terry Francona said Matsuzaka’s progress with the strengthening program, which will be monitored by the team’s athletic training staff, would dictate how soon he progresses to pitching.

“The throwing part will be back according to his strength, probably not the other way around,” Francona said. “We want to make sure he’s real strong and then when the strength is there, he can throw as much as he wants. … We want to not only get that strength, but maintain it.

Meanwhile, outfielder J.D. Drew said he’s feeling better after missing Friday’s game with vertigo symptoms.

Drew said he took medication and kept himself hydrated Friday night, which helped alleviated the lightheadedness that kept him out of the lineup.

“It’s a lot better,” he said. “I’ve had little bouts here and there, but this is the first time it’s been a day or two at a time. It’s never lingered. But I’m 100 percent better one day to the next.”