But where, exactly, they're doing that is up for debate.
According to U.K. newspaper The Telegraph, the Clooneys are unwinding on a two-week $75,000 honeymoon on North Island in Seychelles, the Indian Ocean archipelago. The newspaper reports that Seychelles' tourism minister confirmed the couple's presence there.
Clooney, 53, and Alamuddin, 36, were also recently spotted at Victoria International Airport on the island of Mahé in Seychelles, E! News reports. The couple has previously vacationed on the island chain before they married.
Meanwhile, the Daily Mail is reporting that the Oscar-winning actor and the human-rights attorney returned to the U.K. following their wedding and have taken up residence in a new marital home, a multimillion-dollar mansion in Oxfordshire that borders the Thames River.
Their new home reportedly features a spa complex and home movie theater as well as two cottages on the property, which the paper says is an less than an hour away from Alamuddin's London legal offices and also near Highclere Castle, where "Downton Abbey" is filmed.
Clooney and Alamuddin shared photos of their big day with People, Hello! and Vanity Fair Italy.