Amazon Prime Video agreed to pay as much as 10 million pounds a year for the rights, which include the ATP Tour finals competition between the top eight singles and doubles players, the newspaper said.
Sky was paying about 8 million pounds a year under a deal that lasts until 2018, the Guardian said.
Amazon won the streaming rights in the United States for NFL's Thursday Night Football in April, kicking off a push by the online retailer to attract fans to its shopping and video service.
Sky, Britain's biggest pay-TV group, said last week it had walked away from bidding on some sports rights as it invest elsewhere in the business, including continuing to broaden its drama and entertainment content.
Sky has recently reorganised its sports broadcasting to create channels dedicated to specific sports such as football, cricket, Formula One and golf.
Amazon and Sky declined to comment.
(Reporting by Paul Sandle; Editing by Keith Weir)
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