The Spanish giants announced back in March that they had agreed a deal with Gremio for the transfer of the 21-year-old, and the deal will go through this summer.
Arthur leaves Gremio having scored twice and registered three assists in 50 appearances for the Brazilian club.
"I wanted to leave with a press conference to express the love I have for the Gremio fans," Arthur told reporters. "I'm fulfilling a dream and I have to thank [Gremio] for this.
"It's not a goodbye but rather a 'see you later' and I hope to be able to come back one day. Barca asked for me to move this summer, it was clubs who spoke. I didn't participate in the negotiations.
"I've heard wonders about the club, city and fans, so I have big expectations for my arrival at Barcelona."
Barcelona have reportedly parted with an initial £26.5m for the midfielder, although the deal could eventually reach £34m.