Demiral scored in the third minute of the 2-1 win over Roma at the Stadio Olimpico on Sunday before limping off quarter of an hour later. The 21-year-old fell back on his left leg after jumping for a header.
"Demiral sustained a sprain to his left knee," Juventus said in a statement.
"Surgical intervention will be required in the coming days."
Recovery time for such an injury takes between four and six months.
Turkey play the opening match of Euro 2020 against Italy at the Stadio Olimpico on June 12.
Roma's Italy international Nicolo Zaniolo also suffered a cruciate ligament injury during Sunday's game with his season also at risk.
The 20-year-old was stretchered off the pitch in tears in the 36th minute and will undergo surgery in Rome on Monday.
"Come on Nicolo! I'm waiting for you on the pitch more determined than ever," said Italy coach Roberto Mancini in a Twitter post on Monday.