Kopech is ranked No. 5 among Red Sox prospects by MLBPipeline.com.
"There's a small fracture in his hand," said Red Sox general manager Mike Hazen. "We're still trying to figure out what the medical [recovery] is going to be. We have to go through more tests just to make sure we know exactly what we're dealing with."
The Red Sox are displeased Kopech has cost himself more time away from the mound with another off-field mistake.
"It's disappointing," said Hazen. "Very disappointed. It was stupid. He's going to have to grow up, obviously, with the things that have happened so far. He's got a long road to go to get to the big leagues. He obviously has a ton of potential, and he's got a long way to go. These types of things, you don't want to put more barriers in front of you than playing professional baseball already presents to you."
The Red Sox took Kopech in the first round (33rd overall pick) of the 2014 MLB Draft out of Mount Pleasant (Texas) High School. Kopech is 19 years old.
"I don't want to go into details of what happened," Hazen said. "He was very apologetic. He's very disappointed by what happened certainly, and now he's going to pay for it with regard to more lost time."
In the coming days, the Red Sox should get a better understanding of how much time Kopech will miss.
"It depends on the severity of the fracture and if we have to do anything with it [surgically] or not," said Hazen. "Right now, we're unsure exactly. There's a wide range of time frames depending on what we have to do. We don't have those answers yet."
Kopech pitched in 16 games for Class A Greenville last year, going 4-5 with a 2.63 ERA and notching 70 strikeouts over 65 innings.
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.