The deal would see Vidal reunited with Inter manager Antonio Conte, under whom he won three Serie A titles in a row at Juventus.
Inter's official Twitter account published a photo of Vidal's arrival and a video of the player, 33, saluting fans saying "Hi Inter, I'm here."
Barca captain Lionel Messi paid tribute to his soon-to-be former teammate in a post on Instagram on Sunday.
"I only knew you from playing against you and I always thought you were a phenomenon but after being lucky enough to get to know you personally you surprised me even more.
"We shared many things in the last two years and you made a huge impact on the dressing room and will be dearly missed. I wish you all the best luck in your new club and I'm sure our paths will cross again."
Conte had been keen to seal Vidal's signature a year earlier when he took over at San Siro, but was unable to finalise the deal.
Vidal had won a league title in every season from 2012 until his streak was ended when Barcelona finished behind Real Madrid in La Liga last season.
From 2012 to 2015, Vidal won four Serie A titles with Juve, then he won the Bundesliga in each of his three seasons with Bayern Munich, before he won La Liga in his debut campaign with Barca.