The club said the surgery on the thyroid gland which is located in the neck had been carried out in Frankfurt and “was a complete success”.
“The prognosis for the resumption of physical activity will be better assessed in the coming days,” Juventus added in a statement.
According to reports in the Italian media, the 24-year-old Germany international could be out of action for up to six weeks.
Former Bayern Munich and Bayer Leverkusen player Can joined Juventus on a four-year deal last summer after his contract with Liverpool expired.
He made 167 appearances during four seasons with the Premier League side.