Beckett was not with the team in Florida on Monday after throwing a 100-pitch bullpen session on Sunday.
"He threw 100 pitches yesterday and felt great," Valentine said. "Hit location and he's had a little situation that he's taking care of today ... just for peace of mind."
Beckett is also in line to pitch the Red Sox's home opener at Fenway Park on April 13, against the Rays.
Valentine said he had discretely raised Beckett's thumb issue before.
"[It's] just been a little situation," Valentine said, echoing his description of Bailey's injury. "I think I might've mentioned it 10 days ago that 'There's a situation that we've been dealing with.' As in all cases, you have to be prepared, and I think we are. I think we're covered with whatever happens. The good news is, in his 100 pitches yesterday, he felt terrific."
Pitching coach Bob McClure gave no hint Sunday that Beckett wasn't feeling well when asked how the bullpen session went, saying the righty threw "very well, very good."
Prior to the injury being disclosed, McClure said Beckett was going to throw a side session Tuesday in Washington, where the Red Sox are going to have an exhibition game against the Nationals.
The likelihood now is that Beckett will not meet the team in Washington and instead travel directly to Detroit.
Beckett's injury was revealed Monday morning by another Boston pitcher, Alfredo Aceves, a rotation candidate who was told on Sunday that he was being put in the bullpen. Asked Monday how that conversation with Valentine went, Aceves said that he was told he may have to start in Beckett's place in Detroit and in the home opener.
"[Valentine] said that ... Josh Beckett ... has an injury in his thumb," Aceves said.
A reporter asked Aceves if he meant Bailey had a thumb injury.
"No, Josh Beckett," Aceves said. "If he doesn't start the second game, I will start the second game and the first game at home. That's what [Valentine] said. And then he said, 'But [for] now, you're going to be in the bullpen.'"
Said Valentine of his talk with Aceves: "Well, I'd rather not talk about conversations I have with players. But suffice it to say, you have to always be prepared."
As for Bailey, he was scheduled to be examined in Boston on Monday and possibly again on Tuesday, Valentine said.
Valentine did not shed any further light on who might replace Bailey as Boston's closer if Bailey landed on the disabled list, but Aceves, Mark Melancon or Vicente Padilla could potentially handle the ninth-inning duties.
Valentine gently joked when he was asked if Beckett's thumb affected him more when throwing a particular pitch.
"You'd have to ask him," said Valentine. "It's been a little hard to quite put our thumb on."