Placed on the 15-day disabled list on Sunday with stomach illness, Buchholz remained hospitalized on Monday.
"Last night he was hoping to get out [Monday], but I was just in the training room, and it was decided to keep him a little longer," said manager Bobby Valentine. "Not because anything got worse, but they wanted to make sure it was as stable as possible before he gets out."
The right-hander is suffering from internal bleeding, but the Red Sox have not determined the exact source of the problem.
"I don't think so, no," Valentine said. "I'm concerned, but it's a personal matter with Clay. I bet you he feels OK. He sounded it last night."
Aaron Cook started in Buchholz's place on Sunday vs. the Braves, allowing three runs in five innings to win his first game with Boston.
Beckett, on the 15-day DL with right shoulder inflammation, threw a bullpen session prior to Monday's series opener vs. the Blue Jays. Still, the Red Sox won't place a timetable on his return.
"He said his shoulder felt great," Valentine said. "Because we have him on this disabled list, we're not going to let the calendar dictate. We're going to make sure he's really feeling good. So we'll wait before we make that determination, see how he feels tomorrow."
Beckett did not make himself available to the media on Monday.
He is eligible to return from the DL on Wednesday vs. Toronto, a game Jon Lester is scheduled to start.
Could Beckett start Thursday's game, when Boston begins a seven-game road trip with a four-game series at Seattle?
"I don't think it would be Thursday," Valentine said. "And that, again, is only because of the trip out there [to Seattle]. Why have that as another compounding situation that we have to deal with? Maybe the next day, if everything is going well, but we'll see. He's planning on pitching sometime soon."
Former reliever Franklin Morales has made two starts in Beckett's place, going 1-0 with 17 strikeouts in 11 innings.
Beckett is 4-7 with a 4.14 ERA in 12 starts this season. His last outing was June 11 at Miami, when he allowed four runs over seven innings in a loss.