According to the Mayo Clinic, a subarachnoid hemorrhage is bleeding in the space between the brain and the surrounding membrane.
"I can say that Greg is stable and we have a good prognosis," said Cherington. "Obviously he's in really good hands with the people there at Cleveland Clinic. He will likely be in Cleveland for a few more days, but everything we've heard is there's a very optimistic prognosis and he's doing well.
"Obviously all of our thoughts are with him, his wife, Erica, his family. We know he's getting really good care. Other than that, there's not a lot more I can say, but it's looking good and everyone is optimistic about a full recovery."
Minor League hitting coordinator Tim Hyers is expected to join the Red Sox in Detroit to help assistant hitting coach Victor Rodriguez until Colbrunn returns to the club.
"I'm sure you can ask [manager] John [Farrell] later in terms of division of responsibilities and that sort of thing, but we expect Tim Hyers to join the team and help us out," said Cherington. "And at this point, yes, we do expect Greg to rejoin the team, but it's just too early to say when."
Colbrunn experienced dizziness prior to the start of Wednesday's game and was admitted to the Cleveland Clinic, where it was determined that he had suffered the hemorrhage.
Colbrunn is in his second season as the Red Sox's hitting coach. Boston is batting .244 as a team this year with 45 home runs and a .328 on-base percentage that ranks sixth in the Majors.
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, and follow him on Twitter @IanMBrowne. This story was not subject to the approval of Major League Baseball or its clubs.