With maximum sustained winds of 85 mph, Richard was centred about 95 miles east of Belize City, said the US National Hurricane Center (NHC).
Wind and rain from the 10th storm of the hurricane season have already battered Honduras' Caribbean coast.
Richard is moving north-west at about 12mph. The Miami-based NHC forecasts it will cross Belize and southeastern Mexico later on Sunday, crossing the Yucatan peninsula before reaching the south-western Gulf of Mexico on Tuesday.
Mexican fishermen have already been evacuating the Yucatan peninsula.
A hurricane warning is in effect for the west coasts of Belize and Honduras.
The NHC warned a storm surge would raise water levels along the northern coasts of Belize and Honduras by as much as five feet, adding that this would be accompanied by "large and destructive waves".
Honduras could expect up to two inches of rain.