PORT ROYAL, Jamaica -- Tropical Storm Ernesto brushed past the Caribbean country's southern coast Sunday, causing heavy rains but few problems as it took a course expected to carry it by Honduras and across the Yucatan Peninsula into the Gulf of Mexico.

Forecasters at Jamaica's Meteorological Service said rain was tapering off but light to moderate showers from squalls were still present over most of the island, especially eastern areas.

In Jamaica's southern capital of Kingston, Ernesto dumped some rain but left no noticeable damage.

Island authorities urged people in flood-prone areas to stay on alert and urged fishermen on outlying cays and banks to remain in safe harbours on the main island.

As the storm's outer bands approached earlier, Prime Minister Portia Simpson Miller called on all Jamaicans to make the necessary preparations to ensure their safety.

"I urge you to especially consider the children, the sick, persons with disabilities, the elderly, and the most vulnerable in the society," she said.

Daniel Edwards, a dreadlocked fisherman in Port Royal, a small fishing village at the tip of a spit of land near Kingston's airport, said vigorous lightning lit up the sky over the sea late Saturday and early Sunday.

Bailing out his small wooden fishing boat next to a dilapidated wooden dock, Edwards said he wasn't overly concerned by the storm.

"It's not much of a muchness," said the veteran fisherman, decked out in rain gear.

Forecasters said Ernesto could begin drenching the coast of Honduras on Monday. It is then expected to move ashore near the Belize-Mexico border Tuesday night and pass into the Gulf of Mexico.

The U.S. National Hurricane Center in Miami said Sunday evening that Ernesto was getting a bit more organized as it moved away from Jamaica. Its forecast said the storm might become a hurricane by late in the week while heading toward the Mexican mainland after crossing over Yucatan.

A tropical storm watch was in effect for the coast of Honduras, from the border with Nicaragua westward to Punta Castilla, and the main island of Grand Cayman in the Cayman Islands.

A Cayman government statement urged Grand Cayman residents to monitor the storm but said the storm was not likely to have serious effects on the British Caribbean territory.

The U.S. hurricane centre said Ernesto was centred about 245 miles (395 kilometres) east of Cabo Gracias a Dios on the Nicaragua-Honduras border Sunday evening. It had maximum sustained winds near 50 mph (85 kph) and was moving westward at 20 mph (32 kph).

Elsewhere, Tropical Storm Florence, which formed far out in the Atlantic, stopped strengthening by early Sunday and was no longer expected to gain strength, the hurricane centre said.

Florence's top sustained winds had slowed to 50 mph (85 kph) by Sunday afternoon and it was 845 miles (1,335 kilometres) west of the Cape Verde Islands. Forecasters said a gradual weakening of the storm was expected over the next couple of days.