Son will require surgery on the injury, which happened in a first-minute challenge with Villa's Ezri Konsa, despite going on to complete the match and scoring twice - including a stoppage-time winner in the 3-2 victory.
Tottenham's statement confirming Son's injury stated he was expected to miss "a number of weeks", but Jose Mourinho suggested that was optimistic.
Asked if he expects Son to play again this season, Mourinho said: "I don't. If he plays one game or two games, it's because he (head of communications, Simon Felstein) is very optimistic.
"I hope he's right but in my mind, no. In my mind, I'm not thinking about that.
"If I was the one to write the statement, I would write different. So you can understand what I mean. We miss him."
Son has been their most reliable player since Harry Kane was ruled out, with six goals in his last five games.