Francona said that he will reveal his rotation for the games in Tokyo on Monday. Daisuke Matsuzaka and Jon Lester are the top candidates to pitch the two regular-season games against Oakland on March 25-26.
Sox pitching coach John Farrell, who caught for Beckett on Sunday, said that the right-hander made 25 throws from 60 feet and 10 throws from 75 feet on flat ground with no problems.
Beckett, a 20-game winner a year ago, would have been a lock to pitch Opening Day on March 25. But that went out the window when Beckett had to be removed from a Grapefruit League start on March 8 following just one warmup toss because of back spasms.
"Beckett did fine," Francona told reporters in Bradenton, Fla., before the Sox's game against the Pirates on Sunday.
"[But], saying that, he's not going to go with us to Japan," Francona said. "[I] talked to him this morning about it. He completely understands. In saying that, we'll get to the pitching in the next couple of days.
"It's just not the right thing to do. Everything we say, we need to live by. ... But it was good news today, because he was out there and he wasn't holding his back. John was real excited."
Farrell said Beckett's back gave him no problems during the throwing session. Farrell said he experienced no discomfort and was able to maintain a normal arm action and follow-through on flat ground. The Sox plan to have Beckett throw from 90 feet Monday, then resume his normal long-toss activity Tuesday, with the hopes of Beckett returning to the mound Thursday.
"The one thing about Josh, he's very much in tune with his body," Farrell said. "As good as he felt, he knows there are steps along the way which will we will look to accomplish. The main thing is, even at 75 feet, he got on it a little bit as far as intensity level. I'm sure he wants to go, but he's well aware there will be a throwing program and a plan in place. The fact he feels nothing in there is a clear indication the inflammation has subsided. Now we'll take the appropriate steps to build him up."
Asked if Beckett could possibly start one of the three exhibition games in Los Angeles on March 28-30, Farrell replied: "We'll look to use those games in L.A. to the best of our abilities. Until we get a more clear game schedule, it's premature as to whether he would be out in L.A. at this point."
Farrell thought that the possibility of Beckett pitching one of the two games in Oakland on April 1 or 2, when the Sox resume their regular-season schedule, might be too optimistic.
"I think that's probably a little bit too aggressive at this point, because there's going to be a one-inning progression, much like everyone else goes through Spring Training," Farrell said. "Provided there are no additional setbacks, the first game, when that date is determined, would probably be two innings."
Francona said the Sox are still working on a plan for Beckett while the team is away.
"We're trying to piece those types of things together," Francona said. "We're trying to figure out if [Sox physical therapist] Scott Waugh can come down. We're trying to make sure we have the people in place that know the routine. We talked a little about leaving [rehab coordinator] Mike [Reinold] back, but that doesn't make sense with 14 pitchers in Japan. We'll have it all figured out."
Beckett was not available for comment.
Maureen Mullen is a contributor to MLB.com. MLB.com reporter Ian Browne contributed to this report. This story was not subject to the approval of Major League Baseball or its clubs.