According to Fabrizio Romano, the 26-year-old is now a fully-fledged Juventus player after the obligation to buy was activated.
Kelly joined the Italian giants from Newcastle on loan in January, only six months after he was signed as a free agent from Bournemouth.
Juventus pays a reimbursement of € 17.5 million plus add-on for Kelly, who has made 13 performances in all competitions for the Serie A-side.
Just like Newcastle, Juve is fighting to qualify for next season's Champions League, which is in fourth place in the Italian flight with three more games.