The 35-year-old Portuguese defender arrived in Glasgow on Tuesday for discussions about a move to Ibrox.
The former Porto, Zenit St Petersburg and Fenerbahce centre-back, who has been capped 89 times and who was member of Portugal’s Euro 2016-winning squad, is expected to be Pedro Caixinha’s first signing.
The statement on Cagliari’s official website read: “After a season in red and blue, Bruno Alves leaves Cagliari to face a new professional challenge.
“The club would like to thank Alves for all he has done in Cagliari and wishes now the greatest satisfaction for the continuous of his career.”
Alves has 89 caps for Portugal and is part of their squad for the Confederations Cup in Russia.