The forward left Barcelona's El Prat airport on Thursday lunchtime, headed for Perugia.
Suarez -- who has has agreed terms over a move to Juve, according to ESPN sources -- has known for some weeks that he is not in the plans of new Barca coach Ronald Koeman.
He has been insistient that Barcelona pay him the remaining year of his contract with the club in full before moving on, which they have so far refused to do.
Koeman left Suarez out of the Barca squad for the second game running on Wednesday for a 3-1 friendly win over Girona.
"We are waiting to see if he goes or not, we still don't know," Koeman said after the match. "We respect all contracts here and, if he ends up staying, he will be part of the squad."
Juventus had seemingly withdrawn the offer they made for Suarez weeks ago, and have instead begun talks with Edin Dzeko, although there is no official agreement in place.
Suarez wants to take advantage of the Italian passport of his wife Sofia. To do so, he must pass the exam -- which involves a test of knowledge of Italian culture -- on Thursday.