Arthur will join the Italian champions after completing the 2019/2020 season with Quique Setien’s men.
The two clubs have been locked in conversation over the transfer for a while, and there are reports Bosnian midfielder Miralem Pjanic will move in the opposite direction in a separate transfer.
Arthur joined Barcelona from Brazilian side Gremio in the summer of 2018 and his style of play saw many see him as a long-term successor to Xavi Hernandez.
The 23-year-old made 44 appearances in his debut season and another 28 has been taken in this term.
"FC Barcelona and Juventus FC have reached an agreement for the transfer of player Arthur Melo. The Italian club will pay €72 million, plus €10 million in variables," the club confirmed in a statement published on Monday.
"The player will stay at FC Barcelona until the end of the official competitions of the 2019-20 season."
Barcelona have included Arthur in their squad to face Atletico Madrid on Tuesday as manager Setien is confident he will produce the goods despite his impending exit.
"I have no doubt about his greatest commitment until the last day he is here," the head coach told the media.
"He is a player until the last game of the season. He has committed himself to being one more [member of the squad].
"He will dedicate himself fully to achieving the objectives. I have no doubts about him.
"He has said that he will be in the best conditions. I'll decide if he plays more or less."
Arthur has won one La Liga title and one Spanish Super Cup during his time in Spain.
