Smith, who was set to share closer duties with Jeremy Jeffress, received the diagnosis Friday night and was awaiting test results to determine if surgery is necessary.
Smith said he was getting ready to shower after pitching in a minor league game Thursday and was standing on one leg to take off his other shoe when he lost his balance and twisted the knee.
"I pulled hard [on the shoe] and it stayed on," he said. "My knee just went up and popped. Everyone tells you there is nothing you can do about it, but you still feel like you are letting people down."
Smith tore his lateral collateral ligament, which is on the outside of the knee and connects the femur to the lower leg. The right leg is the plant leg for the left-hander and absorbs the most impact.
Manager Craig Counsell said Smith will miss "significant time," and the Brewers are expected to find out exactly how long when he is examined by team physician Dr. William Raasch in the coming days.
"The recovery is a little longer with surgery," said Counsell, who said it is not a season-ending injury. "It's tough for us, it's tough for Will. There is going to be somebody in [the clubhouse] that takes this opportunity and makes a lot of it."
Smith, who was 7-2 with a 2.70 ERA in 76 appearances last season, was having an outstanding spring. He had allowed only two hits in seven innings, striking out five. Counsell named Smith and Jeffress, good friends and roommates this spring, as the co-closers last week.