The announcement was made by general manager Steve Yzerman after Saturday's 3-1 win over the New Jersey Devils. The surgery will take place in Tampa, Florida, and Stamkos still plans to return during the playoffs.
The forward played in Thursday's game against the Canadiens, but sat out Saturday night with what was called an upper-body injury. Yzerman called his condition a type of vascular thoracic outlet syndrome near his collarbone.
The only other major injury Stamkos has dealt with in his career was a broken right tibia, which caused him to miss 45 games of the 2013-14 season. The team is already without Anton Stralman, who will miss at least the first round due to a fractured left leg.
Tampa Bay will not know the exact timetable of Stamkos's absence for another couple of weeks.
The Lightning have not announced an official replacement for their star, but they do plan on calling Jonathan Drouin up form the AHL. Drouin scored twice in 19 games before getting sent down, but has nine goals in his past 10 games for the Crunch.
The Lightning have yet to clinch a playoff spot, but sit in second place in the Atlantic Division with 95 points, two points behind the Florida Panthers.