Kjaer injured his left knee during the early stages of Milan's 3-0 win against Genoa on Thursday (AEDT) and was taken off on a stretcher.
Milan announced on its official website that the Denmark international underwent surgery on his anterior cruciate ligament on Friday and is expected to sit out the next six months.
Speaking earlier in the day ahead of Sunday's (AEDT) clash with Salernitana, head coach Pioli confirmed Milan would look to bring in cover should Kjaer's injury be long term, which it has proved to be.
"It's clear that I believe an intervention will be needed if that's the case," Pioli said. "But we will only do it to try to improve the team.
"It's worth changing to improve the team. The club, if necessary, will be ready."
Kjaer has been a key player for Milan since joining from Sevilla in January 2020 on an initial loan deal and has started 13 of its 20 matches in all competitions this season.
The 32-year-old, who finished 18th in this week's Ballon d'Or voting, leads the way for interceptions (66) among Milan defenders in Serie A since making his debut.
Milan's first match without Kjaer will be at home to bottom side Salernitana this weekend in what is the first top-flight meeting between the sides since January 1999.
The Rossoneri have lost only one of their past 20 Serie A games against newly promoted sides, keeping 12 clean sheets across that period.
With Milan one point adrift of leader Napoli, Pioli's side has a chance to build some momentum with victory over Salernitana.
"Tomorrow's match is important to give continuity," Pioli said. "We have another opportunity to demonstrate that we are a strong team.
"The opponents are tough, very annoying in the non-possession phase and come at you with aggression and conviction. It's a game that we want to do well in."