"I'm finally at Inter and I'm thrilled to be part of this great club," he told Inter's television channel.
Asked if it is true that he is nicknamed the Pitbull, Medel replied: "Yes, it is ... I bite."
Italian media said the transfer was finalised on Saturday but it was not confirmed by Inter at the time.
Medel, who can feature in the centre of the defence or midfield, was one of the outstanding players as Chile reached the last 16 of the 2014 World Cup, beating title holders Spain on the way.
The 27-year-old, famous for his extraordinary will to win, spent one year at Cardiff having previously played for Universidad Catolica, Boca Juniors and Sevilla.
Raised in one of Santiago's toughest neighbourhoods, Medel has recently put behind him a poor disciplinary record which saw him receive 13 red cards and become involved in a series of controversial incidents.
In September 2007 he was given a six-match ban for spitting at an opponent during a Chilean league match against Deportes Concepcion and one month after that he was arrested for drink-driving.
In January 2009 Medel was injured when his car crashed on the highway from Vina del Mar to Santiago and he was thrown out of the vehicle.
Recalling the incident later, he said that as he sat slumped in the wreckage of his vehicle he could not feel his legs and feared he was paralysed from the waist down.
A month later a teenage girl fell to her death from the ninth-floor balcony of his apartment. An investigation cleared him of wrongdoing but the incident cast a shadow over the player.
Medel once said that the sport was his lifeline. "For my own sake it's a good job that I chose football," he explained. "If not perhaps I'd be stealing or drug trafficking."
(Writing by Brian Homewood, editing by Tony Jimenez)
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