The 23-year-old Bjorkstrand, the team's fifth-leading scorer last season with 11 goals and 29 assists, signed a three-year pact worth $7.5 million.
The 24-year-old Murray accepted the team's qualifying offer of a one-year contract worth $2.8 million. Both deals were announced over the weekend.
The Blue Jackets' top scorer Artemi Panarin tweeted that he and his agent were meeting for contract talks with team general manager Jarmo Kekalainen on Monday in France. He is a free agent after next season.