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The 31-year-old, who joined Chelsea from Napoli in 2018, was in the final six months of his contract at Stamford Bridge.
He passed a medical on Tuesday and signed an 18-month contract with the option of a further year.
The Gunners turned their attentions to Jorginho after having two bids rejected by Brighton for Ecuador midfielder Moises Caicedo.
Brighton said Caicedo was not for sale and he looks to set to remain at the club after the 11pm deadline.
