Danielle became the second hurricane of the Atlantic season on Monday with winds reaching sustained speeds of 85 miles (140 kilometres) per hour, the US National Hurricane Center said.
"Danielle is a category one hurricane on the Saffir-Simpson hurricane wind-scale," the centre said in an advisory.
"Additional strengthening is forecast during the next 48 hours and Danielle is forecast to become a major hurricane by late Tuesday.
But the forecast track for the storm show it passing east of Bermuda and turning north-northeast to miss the US east coast. The centre said it posed no land hazards at present.
At 0300 GMT, the storm was about 1,205 miles (1,940 kilometres) east of the Lesser Antilles, moving in a west-northwesterly direction at a speed of 20 miles per hour (32 kilometres).
"Maximum sustained winds are near 85 mph (140 kilometres per hour) with higher gusts," the hurricane centre said.
The first hurricane of the Atlantic season was Alex, which left one person dead in northern Mexico and disrupted oil clean up efforts in the Gulf of Mexico before being downgraded to a tropical storm on June 30.
Forecasters are also tracking Tropical Storm Frank, a Pacific weather system expected to bring high winds and rains to Mexico's western coast.
Follow Emirates 24|7 on Google News.
