Neto, 29, joins Barca for an initial fee of €26 million, with an additional €9m due in potential add-ons. He has signed a four-year contract at Camp Nou, tying him to the Spanish champions until 2023, and has had a €200m buyout clause inserted into his terms.
Cillessen signed for Valencia for an up front fee of €35m earlier this week. If Barca end up paying all the add-ons for Neto, the deal between the clubs will end up being a de-facto player exchange.
Barca were keen for Cillessen to remain at the club but the Dutch goalkeeper insisted on a move. He told them several months ago that his desire was to leave this summer so he could play regular football ahead of next year's European Championships.
Neto made 80 appearances for Valencia during his two seasons with the club after joining from Juventus, where he was back up to Gianluigi Buffon, in 2017. He has also played for Fiorentina and Atletico Paranaense and has one cap for Brazil's national team.
He will be No.2 at Barca to Germany international Ter Stegen, who has kept Cillessen out of the team for the last three seasons.