The team said X-rays at the arena revealed the fracture and that Bogut was taken to the Cleveland Clinic for further evaluation.
"It's very deflating ... a tough moment," LeBron James said. "We all were excited about the acquisition and bringing him in here.
"We can hope for the best ... but it's a tough one not only for him, first of all, but for our ballclub."
James said he knew right away what had happened.
"As soon as the collision happened, I heard it break," he said. "And when I went over to him and he said [it was broken], I already knew it. I heard it crack.
"It took the air out of the whole building."
The injury occurred when Bogut was closing out on Miami Heat forward Okaro White and his left leg collided with White's left knee with 11:38 remaining in the second quarter. Bogut had played just 58 seconds.
He was helped to the locker room by Tristan Thompson and James Jones, and did not attempt to put any weight on the leg.
"When you bring a guy in, you don't expect that type of thing happening a minute after he checks into the game," said Cavaliers assistant coach Larry Drew, who filled in for the ill Tyronn Lue in Cleveland's 106-98 loss.
"It can certainly take a little bit out of your sail. But injuries are a part of it, and you have to play through it.
"All we can do is wish the best for him."
Drew said he did not have a chance to speak with Bogut before he left for the hospital for further evaluation.
Kyrie Irving said the Cavs still have enough to compete for a championship even without Bogut.
"[General manager] David Griffin has done an unbelievable job, and the pieces that we do have, I mean, this is probably the best group that I've played on since I've been here," Irving said. "And that's hands down. And talent all around. So I'm definitely still confident about it, and I always will be."
Earlier Monday, the focus was on what Bogut's addition could mean for a possible third installment of Cavaliers-Golden State Warriors in the NBA Finals come June.
The new Cavs center -- and former Warriors big man -- said his old team had nothing to do with his decision to join his current one.
"I mean, we'll play whoever's in front of us at this point," Bogut said before the Cavs played the Heat. "That's the mindset. Everyone's going to try to ramp up that story -- nice try -- it's one of those things. Happened to [Anderson] Varejao last season. That's the business of this league and pro sports. I don't hold any grudges and moved on and I'm in a good place, and we'll see how it goes."
Bogut signed with the Cavs last week after being courted by the likes of the Boston Celtics (whom Bogut identified as the runner-up), Houston Rockets and San Antonio Spurs once he reached a buyout agreement with the Philadelphia 76ers following his trade from the Dallas Mavericks.
He started this game of musical chairs with his career the summer after Golden State traded him to Dallas following four years of service and two trips to the Finals in order to clear the requisite cap space to sign Kevin Durant in free agency.
"You always say you want to get back at them, this that, this that," Bogut said of the Warriors. "You know, it's nothing to do with them. This decision was solely made on, you know -- I wasn't making this decision based on Golden State. I was making this decision based on what the best opportunity was. I'm a free agent at season's [end], and a chance to get another ring would be absolutely awesome."
What wasn't so awesome was the red tape Bogut had to contend with on Monday as he was held out of Cavs shootaround while he and the team tried to clear up an issue with transferring his work visa.
"Think it's just when you change jobs; my visa's based on a job," said Bogut, who is from Australia. "So if I'm working at U.S. Bank or McDonald's, that's what it says on my visa. So when you change teams, they have to bring on your visa rights, removing cities, immigration wants to know why you're moving cities, so you've got to be able to back that up with a contract from your company. I thought it was all done, but it was obviously the weekend. I came to the facility and they were like, 'You can't go right now, you have immigration issues.' So I called Vladimir Putin and got it done."
While Bogut made a joke about the state of international diplomacy, the Cavs played a joke on him for his first day with the team. The former No. 1 pick out of Utah has worn No. 6 for the majority of his 12-year career, but No. 6 is what Jameswore in Miami and what he often still wears on his practice jersey. So, Bogut's practice jersey was presented to him with the No. 6.5, as an ode to James. (Bogut will wear No. 6 on his game uniform, however.)
"LeBron got me already," Bogut said with a smile. "I wasn't aware that LeBron wears a No. 6 at practice, so they gave me a 6.5. I thought it was 65 at first. I didn't really pay any attention to it. It was early in the morning, I just got to Cleveland a couple days ago, so I'm still a bit weary. Someone mentioned the point. It was a good joke. I have to get him back."
Bogut, 32, who averaged 3 points and 8.3 rebounds per game for Dallas this season, said he dealt with "crap I caught on Twitter and the death threats and all that kind of stuff" for choosing the Cavs, but he sounded resolute in his choice nonetheless.
"It's the best opportunity to try to win a championship playing with one of the best -- if not the best -- players to play the game," Bogut said of James. "It was a tough decision, in a way, because of the history, but I can't control it. I got traded so I ended up having the luxury of having four or five teams recruit me, which is a nice thing to have. But at the end of the day, I thought Cleveland was the best chance."