The team said Thursday that Koivu will have the operations in his native Finland in the next couple of days.
Koivu will have arthroscopic surgery to repair a torn labrum in the shoulder, requiring 14 to 16 weeks of rehab. He will also have arthroscopic surgery on the knee, requiring three to four weeks of recovery.
The team captain had a career-high 22 goals and 49 assists in 80 games for the Wild last season.
The Wild also say defenseman Cam Barker had arthroscopic surgery to remove loose particles from his right ankle on Tuesday. He will need three to four weeks of rehabilitation.