The Rangers extended Bergman a qualifying offer Tuesday.
On Monday, the Blue Jackets traded Bergman, along with a pair of 2019 draft picks, to the Rangers in exchange for Adam McQuaid, Kevin Weekes of NHL.com reports.
Bergman was sent to Columbus along with star Matt Duchene, with the Senators getting back prospects Vitaly Abramov and Jonathan Davidsson, Bob McKenzie of TSN reports.
Bergman was moved to Ottawa on Tuesday as part of the deal that saw Mike Hoffman join the Sharks.
The Sharks have cut Bergman from training camp, the team announced Monday.