The left-back has been heavily linked with a switch to Stamford Bridge as Antonio Conte seeks to strengthen the Premier League champions.
Having already lost Leonardo Bonucci and Dani Alves from his highly-regarded defence, Allegri categorically denied the 26-year-old will be following them through the exit door.
"Alex Sandro is a Juventus player and he will stay one," the Italian said at Wembley.
One player who does look to be leaving the Serie A champions is 23-year-old Gabon international Mario Lemina.
Southampton are widely reported to have agreed a fee for the midfielder and Allegri admitted discussions are under way with potential suitors.
"We will see what happens. We are in talks with several clubs," he said. "For the moment Lemina is still a Juventus player."