According to the National Hurricane Centre (NHC), this strengthening trend should begin to ease soon.
Bermuda remains under Tropical Storm Watch.
The NHC has advised that tropical storm conditions are possible within the watch area within 36 hours.
At 5:00am AST (0900 UTC), the centre of Hurricane Epsilon was located near latitude 28.9 North, longitude 58.3 West.
Epsilon is moving toward the west-northwest near 14 mph (22 km/h), and this motion at a slower forward speed is expected through today.
A turn toward the northwest is expected tonight, followed by a turn toward the north by Thursday night.
On the forecast track, the centre of Epsilon is forecast to make its closest approach to Bermuda Thursday afternoon or evening.
Maximum sustained winds have increased to near 85 mph (140 km/h) with higher gusts.
Some additional strengthening is possible today, followed by little change in strength into the weekend.
Hurricane-force winds extend outward up to 25 miles (35 km) from the centre and tropical-storm-force winds extend outward up to 435 miles (705 km) mainly to the north of the centre.
The estimated minimum central pressure is 976 mb (28.82 inches).
Large swells generated by Epsilon are affecting Bermuda, The Bahamas, the Greater Antilles, and the Leeward Islands, and are expected to reach portions of the east coast of the United States and Atlantic Canada during the next couple of days.
These swells are likely to cause life-threatening surf and rip current conditions.