Epsilon, named after the fifth letter in the Greek alphabet, was around 780 miles (1,255 km) east of the wealthy British island territory by 10 a.m. EST/1500 GMT, the US National Hurricane Center said. The center said the storm had been pounding Bermuda with heavy surf for a couple of days.
The storm bore maximum sustained winds of 65 mph (105 kph) and could strengthen slightly before weakening over cooler waters on Friday. It was heading toward the northeast at 10 mph (17 kph).
The 2005 Atlantic hurricane season, which officially ended on Wednesday, has gone into the record books for the most storms and the most hurricanes.
It also had the most Category 5 storms - the top rank on the five-step Saffir-Simpson scale of hurricane intensity - including Hurricane Katrina, which devastated New Orleans and killed more than 1,200 people in Louisiana and Mississippi.