The Spaniard joined the Italian champions for around €20 million last month from Real Madrid but was ruled out for 50 days just a few days later.
However, the 21-year-old believes he is recovering well and could be back in just over four weeks.
"My knee is getting better, I'm recovering well," he told AS. "In about 30 or 35 days I will be ready.
"It was a shame about the injury but I'm working with the physios to play again soon.
"Signing for Juventus has been a great joy, I'm really happy with my new team."
Real Madrid have retained a buyback option in the terms of Morata's sale and he could return to his boyhood club for around €30m.
Juventus are currently on a pre-season tour of Asia and the Pacific.