Amazon Prime Video agreed to pay as much as 10 million pounds ($13.2 million) 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.