The Ferrari website said the 27-year-old met chairman Sergio Marchionne and new team principal Maurizio Arrivabene on Monday after talks with engineers and time in the simulator over the weekend.
On Saturday, he completed nearly 100 laps of the Fiorano test circuit -- a place he visited many years ago as a fan hoping to get a glimpse of his hero Michael Schumacher at work.
"Looking back to the years when I was 11 or 12, trying to look over the fence and trying to see a glimpse of Michael running on the track, well today I was the one running," said Vettel.
"There are a lot of fairytales about Ferrari and how it feels to drive a red car and in the end I can only confirm these fairytales," he added.
"It's not just a story, it's a true legend that exists and it feels really, really special to become part of that and to be inside the car, to see the people coming, running to the track and trying to climb the walls to see the car."
Vettel, who has agreed a three-year contract, replaces Spaniard Fernando Alonso at Ferrari after the double champion made it clear he wanted to leave.
Ferrari finished a dismal fourth overall this season, their first since 1993 without a race victory.
(Reporting by Alan Baldwin, editing by Tony Jimenez)
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