Bergeron had 65 points (25 goals, 40 assists), including scoring his 400th goal, in 73 games last season and won the Selke Trophy as the best defensive forward for the fifth time, an NHL record.
Jim Montgomery, who was introduced as Bruins coach on Monday, said the first call he made to a player after getting the job was Bergeron.
"It was really good. It was enlightening. I felt like I was talking to a coach, not a player," Montgomery said. "Just how he thinks about the team first, thinks about ways to get better. So, very impressive. It was a very open and honest conversation about how he's excited about what the Bruins family can do and how I'm excited to be a part of it."
Krejci, 36, spent his first 15 seasons in Boston but did not play in the NHL last season after opting to return to his native Czech Republic to be closer to his family. There, he played for Olomouc HC and had 46 points (20 goals, 26 assists) in 51 games, but he has never closed the door to a return to the NHL.
Sweeney said the Bruins would work hard to fit him into their roster but that it would be difficult given the limitations of the NHL salary cap, which is set at $82.5 million next season.
"I've had numerous conversations with David Krejci about what his plans are," Sweeney said. "He's spent a lot of time with his family as well, and his decision could be forthcoming. We're hopeful that it's with us. We would welcome him back."