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According to a website X17Online.com, the couple is staying at a luxurious resort on the island, which can only be accessed by helicopter, reports dailymail.co.uk.
Clooney and Alamuddin exchanged vows at a ceremony in Venice Sep 27.
Clooney's close friend and actor Brad Pitt recommended the private getaway for the duo's honeymoon.
"George and Amal were looking for the ultimate, private getaway and I believe it was Brad who told him to go to this resort," said a source.
"It's one of the most beautiful places on earth and George wanted to show Amal something even more amazing than Venice. It was the only place that could top off such an elaborate wedding," the source added.