The defensemen was injured in the first period a 2-1 loss at the Tampa Bay Lightning on Thursday.
"We don't know the exact timeline but somewhere around that four-week range," Senators coach D.J. Smith said Saturday. "Which hurts us. Hurts us defensively. I think he was really coming and playing probably as good as he's played, him and [defenseman Thomas Chabot] really were meshing together there."
Zaitsev scored two points (one goal, one assist) and averaged 18:32 of ice time in 22 games. He was playing on the top defense pair with Chabot.
The Senators complete a three-game road trip at the Philadelphia Flyers on Saturday (7 p.m. ET; SN360, SN1, CITY, NBCSP, ESPN+, NHL LIVE).
Smith said Artem Zub could start on a pair with Chabot, but that the pairs likely will change throughout the game.
Jacob Bernard-Docker, who has been a healthy scratch for three games since being recalled from Belleville of the American Hockey League on Dec. 9, will play his first NHL game since Nov. 14.
Smith said he's confident that a week's worth of practicing and being around NHL players will be beneficial for Bernard-Docker.
"He's a smart player, does a bit of everything," Smith said. "Obviously offensively it's going to take a bit longer at the NHL level. But he can defend. He's a smart kid.
"He's up to NHL speed now, he's had lots of practices. But the difference between American Hockey League and the NHL is a huge jump. And we have to be aware of that and not put too much stock into one game or two games or three games. This is a process for these guys."
