Melissa is about 180km south of Kingston and expected to make landfall in Jamaica on Monday night or Tuesday morning local time (Tuesday afternoon or overnight AEDT), before hitting southeastern Cuba, the US National Hurricane Center (NHC) said.
Catastrophic floods, landslides and storm surges are expected in the region.
"Conditions (in Jamaica) are going to go down rapidly today," the centre's deputy director Jamie Rhome said.
"Be ready to ride this out for several days."
During a press conference in Kingston, authorities said both international airports are closed and 881 shelters have been activated around the nation.
"Many of these communities will not survive the flooding," Desmond McKenzie, minister of local government, said at the press conference.
"Kingston is extremely low. No community in Kingston is immune."
Authorities said a decision on whether to issue a mandatory evacuation order would be made later in the day, while imploring residents to evacuate, with assistance being provided.
Some residents, however, chose to stay put, and authorities said some have refused to board evacuation buses.
"We are riding out the storm, we're not going to go anywhere. Just sitting down and watch everything, make sure everything's working good," said Douglas Butler, a local boat captain in Port Royal.
"I just take things easy. As long as I have my food, I can eat. That is the most important thing."
The NHC said Melissa is expected to bring 40 to 76 centimetres of rain to Jamaica and southern Hispaniola with a local maximum of up to 100cm. Eastern Cuba is expected to receive 25 to 40cm, with local amounts of up to 50cm.
Jamaica's information minister Dana Morris Dixon said the Caribbean Disaster Emergency Management Agency is ready to provide assistance, and several international partners have already pledged support.
"We will experience the intensity of the hurricane tomorrow night Monday going into Tuesday," Morris Dixon said.
"This is rainfall we've not seen before."
The erratic and slow-moving storm has killed at least three people in Haiti and a fourth person in the Dominican Republic, where another person remains missing.
Haitian authorities said three people had died as a consequence of the hurricane and another five were injured due to a collapsed wall.
There were also reports of rising river levels, flooding and a bridge destroyed due to breached riverbanks in Sainte-Suzanne, in the northeast.
Many residents are still reluctant to leave their homes, Haitian officials said.
The storm damaged nearly 200 homes in the Dominican Republic and knocked out water supply systems, affecting more than half a million customers.
It also downed trees and traffic lights, unleashed a couple of small landslides and left more than two dozen communities isolated by floodwaters.
With AP
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