“Everybody in Jamaica needs to be in their safe place and be prepared to stay there for at least the next 12 hours,” Dr Brennan warned.
The BBC’s Nick Davis said Jamaicans had been rushing to supermarkets on Tuesday to get “as much as they could as quickly as they could”.
Jamaica’s Information Minister Dana Morris Dixon said the island had 900 shelters to house people who needed to leave their homes.
In Venezuela, Hurricane Beryl brought heavy rains which caused a river to overflow in the northern state of Sucre. Three people died and several are still missing.
A government delegation was hit by a falling tree while inspecting damage.
President Nicolás Maduro said Vice-President Delcy Rodríguez was among those injured. He said she was “very bruised but conscious”.
In Mexico, where Hurricane Beryl is expected in the coming days, residents in Cancún have rushed to supermarkets to stock up on supplies. Some have encountered empty shelves.
The NHC said that Hurricane Beryl was the earliest category five storm recorded in the Atlantic and had formed much earlier in the hurricane season than usual.
Meteorologists have also remarked on how quickly Beryl developed.
The storm strengthened from a tropical depression into a major hurricane in 42 hours, hurricane expert Sam Lillo told the Associated Press news agency.
In Texas, officials warned residents to prepare for the possibility of Beryl’s arrival this weekend.
On Tuesday, Governor Greg Abbott told resident’s near the state’s Atlantic coast to “keep an eye on the gulf” and “have an emergency plan to take care of yourself and your loved ones”.
The US National Oceanic and Atmospheric Administration (NOAA) has warned that the North Atlantic could get as many as seven major hurricanes this year – up from an average of three in a season.