Vidal is set to be confirmed as a Nerazzurri player on Monday after the buying club posted updates on his arrival in Italy this weekend.
The 33-year-old, who previously played in Serie A for Juventus, is set for a €1million move after two years at Barca.
Former Bayern Munich man Vidal has secured his exit, while Suarez's situation remains unclear, after Messi failed in his attempts to force through a transfer of his own.
As well as Manchester City, Messi – who wanted to be released from his contract – was linked with Vidal's new employers Inter.
He wrote to the Chilean on Instagram on Sunday: "I only knew you from facing each other and you always seemed like a phenomenon to me, but later I was lucky to meet you personally and you surprised me even more.
Te conocía sólo de enfrentarnos y siempre me pareciste un fenómeno, pero después tuve la suerte de conocerte en lo personal y me sorprendiste más todavía. Fueron dos años compartiendo muchas cosas y te hiciste notar mucho, el vestuario te va a extrañar, @kingarturo23oficial. Te deseo todo lo mejor en esta nueva etapa en tu nuevo club. Nos volveremos a cruzar, seguro. 👍🏻💪🏻
"They were two years sharing many things and you made yourself noticed. The dressing room is going to miss you.
"I wish you all the best in this new stage at your new club. We will cross paths again, for sure."
Vidal commented: "Thanks alien! I am proud to have played with the greatest in history and thank you very much for your friendship. I miss you. See you soon."
Vidal was part of the 2018-19 LaLiga title-winning team at Camp Nou and scored 11 goals in 66 league games in total.