The 33-year-old German replaces Mexican Sergio Perez at the team formerly known as Racing Point.
Ferrari drivers Charles Leclerc and Sebastian Vettel have been seen mixing with people outside their individual team groups.
The two-time Le Mans 24 Hours endurance winner and former Ferrari driver, who turns 39 this month, secured his championships in 2005 and 2006 and most recently raced for McLaren in 2018.
Australian Daniel Ricciardo will leave Renault and race for McLaren in 2021.
German media, who flagged up the departure on Monday night, reported Vettel had rejected the terms being offered by Ferrari.
Ferrari have said Vettel remains their first choice to partner young Monegasque Charles Leclerc but reports in Italy have suggested they want an answer by the end of April to ensure potential replacements are still available.
The 34-year-old Briton finished second at the Circuit of the Americas to become only the second driver after retired Ferrari great and seven-times champion Michael Schumacher to win six titles.
The German, whose last victory was in Belgium in August 2018, took the chequered flag 2.641 seconds ahead of unhappy Monegasque team mate Charles Leclerc.
Vettel was demoted to second after being slapped with a five-second penalty for a dangerous re-entry after running off the track.
The Ferrari driver exuded confidence he can overhaul championship leader Lewis Hamilton in the seven remaining races if he and the team avoid making any major mistakes.