Davies is due to undergo surgery on Wednesday and will be out for at least six months. Upamecano will not be available to the club for "several weeks", the Munich club said.
This means the duo will also be out of the Champions League clash with Inter Milan for the time being.
"Unfortunately, there is always the risk of players coming back injured during international breaks - this time it has hit us particularly hard," said sporting director Max Eberl, emphasising: "The absence of Alphonso Davies and Dayot Upamecano weighs heavily on FC Bayern."
He promised Davies "all support" on his way back and added: "We will also be closely monitoring Upa in the coming weeks and assume that he will be available to us again soon.
The Munich squad is "strong and will make up for these absences. We will now move even closer together. The quality is there to continue pursuing our big goals."
Canadian had to leave the pitch early
Davies went down early in the match for third place in the CONCACAF Nations League against the USA, holding his right knee. He was substituted in the 12th minute and returned to the sidelines after brief treatment with an ice pack on his knee.
Munich CEO Jan-Christian Dreesen had given the all-clear after receiving positive feedback from Canada. Back in Munich, Davies was thoroughly examined and a diagnosis was made.
Upamecano was last in action with France. Last Sunday, he converted the decisive penalty in the Nations League quarter-final second leg to win 5-4 against Croatia.