They expect him to be a top player in the long-term. Where they see him fitting in the short-term, especially next season, is still unknown.
Teravainen made his NHL debut in March after finishing among the points leaders in the Liiga in Finland and played in three regular-season games for the Blackhawks before Bowman and Quenneville decided he wouldn't be involved in the team's playoff push. He was reassigned to the AHL and completed his season there.
Bowman has considered Teravainen a steal since the Blackhawks selected him at No. 18 overall in the 2012 draft, and he has used a wide range of adjectives to describe Teravainen's elite offensive game and overall talent. But since recalling Teravainen from Finland became a realistic possibility in March, Bowman has danced somewhere between still calling Teravainen a can't-miss prospect and telling fans not to expect him to be a game-changer the moment his skates land on the ice.
That was Bowman's message again Tuesday.
"We're very excited about Teuvo, but I can tell you we're not going to put any pressure on him," Bowman said. "He's 19 years old still. The one thing we have done very well over the past few years is we've given players time to develop. We haven't put players in positions to fail. And we're not going to do that with Teuvo.
"He's a special player. I've got very high hopes for him, but the one thing I don't have for him is a timeline. He doesn't have to be in any position by any certain date. That's going to take care of itself over time. If anything, we're going to be more cautious like we've been with a lot of our players, giving them time to develop. He's a special player, and he's going to be one for a long time. You look at where he's going to be in five years, we're excited about that."
The key phrase to take from Bowman is "haven't put players in position to fail." The Blackhawks have a recent history of not rushing prospects while giving them time to develop. That can be seen in their current NHL roster with players such as Corey Crawford, Bryan Bickell, Ben Smith and Jeremy Morin.
At the same time, the Blackhawks also have a recent history of not holding back young players when the organization believes they're ready. Jonathan Toews, Patrick Kane, Nick Leddy, Brandon Saad and Andrew Shaw all became permanent lineup fixtures at 20 years old or younger. Teravainen will turn 20 in September.
If Teravainen proves to Bowman and Quenneville he's also ready during training camp this season, there's little reason to believe Teravainen won't be in the team's lineup come opening night. The Blackhawks have a need next season for a center with Michal Handzus becoming an unrestricted free agent. Teravainen is the logical replacement, especially after Bowman traded center Brandon Pirri during the season.
Quenneville shared Tuesday what he learned about Teravainen during his short stint with the team this season.
"He got a short audition here for a couple of games," Quenneville said. "You know, a long year when he comes in and gets meaningful teams, meaningful games. Our games were kind of an evaluation stage. I don't know if that's the right word, but maybe in his presence.
"And I thought, well, it's big league and a strong league. You know, I think he's got skills to play here, and I think as you go along here, I think he's going to have to find a way to get stronger and playing away from the puck, I think he'll adapt quickly to that. I think offensively he's going to have to go in hard areas, and I think that's a process where he'll probably get more comfortable as you go along here and that strength will be something that you build along with.
"But I think his instincts are high-end, that the adaptation process will be quick for him and not putting a timeline on when he'll start. But certainly the opportunity, looking forward for him down the road, is exciting."
Teravainen understood the Blackhawks' stance on him this season. He was happy to make his NHL debut, get a few games in, practice with the team and end his season getting more ice time in the AHL. He wants more next season, though.
"I think I can play in that league right now," Teravainen said in April. "I have played with men in Finland for two to three years, so I'm ready to play there. I think I played OK, but I think it's not enough. I can play there, but it'll take some time getting used to this league and this North American hockey. It's different than European hockey.
"That's my goal to be with the Blackhawks next season, when the season starts. I just try to work hard in the summer. That's going to be a lot of work."