The eye of Dorian, a category one hurricane, has missed the main island of Puerto Rico.
The US territory's smaller islands, Vieques and Culetra, have been battered by heavy rain and high winds.
Dorian is considerably less powerful than the hurricanes that hit Puerto Rico two years ago.
It is expected to gather strength in the coming days, and is on course to reach the US states of Florida or Georgia by the weekend.
The National Hurricane Center (NHC) on Thursday said no watches or warnings were currently in effect for land areas as the storm moves into open water, but tropical storm force winds could begin in parts of Florida by Saturday evening.
Florida declared a state of emergency on Wednesday. Governor Ron DeSantis said it was "important for Floridians on the East Coast to monitor this storm closely".
However, experts say it is too early to say exactly where or when it could arrive.
Dennis Feltgen, a NHC meteorologist in Miami, told news agency AP that the "large storm" could land anywhere from South Florida to South Carolina.
What's the forecast?
As of Thursday morning, the storm is packing winds of more than 85mph (140 km/h) and could bring up to 12in (30cm) of rain when it reaches land. It is currently some 90 miles (145km) north of Puerto Rico's capital, San Juan.
The NHC expects Dorian to reach major hurricane status by Friday.
Forecasters have warned of life-threatening flash flooding and rip-current conditions as the storm moves across the region.
On Wednesday, wind gusts of 111mph (178 km/h) were reported close to St Thomas in the US Virgin Islands, just east of Puerto Rico.
The storm's first victim was reportedly an 80-year-old man, who police said died after falling off a ladder as he tried to prepare for the hurricane in San Juan.
Governor Wanda Vazquez Garced had enacted a law banning the sale of alcohol for at least the next 24 hours, according to CNN. Two cruise liners adjusted their itineraries to avoid the territory this week.
Schools were closed as the storm passed, but will be re-opening on Thursday, CNN reported.
Some 23,000 people were said to be without power in Puerto Rico by Wednesday afternoon, with further power cuts reported in the US Virgin Islands, AP said.
According to Mr Feltgen, the storm is projected to reach the US on Sunday or Monday.
What did President Trump say?
Late on Tuesday, President Trump approved an emergency declaration authorising federal agencies to provide disaster relief.
He also lashed out at Puerto Rico as the island hunkered down for Dorian's approach.
In a series of tweets on Wednesday, he said the island's government was "broken" and "corrupt".
He added that he was "the best thing that's ever happened to Puerto Rico".
Puerto Rico is one of the most corrupt places on earth. Their political system is broken and their politicians are either Incompetent or Corrupt. Congress approved Billions of Dollars last time, more than anyplace else has ever gotten, and it is sent to Crooked Pols. No good!....— Donald J. Trump (@realDonaldTrump) August 28, 2019
....And by the way, I’m the best thing that’s ever happened to Puerto Rico!— Donald J. Trump (@realDonaldTrump) August 28, 2019
Carmen Yulin Cruz. mayor of the Puerto Rican capital San Juan, tweeted: "THIS IS NOT ABOUT POLITICS; THIS IS ABOUT SAVING LIVES."
I said yesterday that Trump needs to be quite, “calm down” get out of the way and make way for those of us who are actually doing the work on the ground. Maybe Trump will understand this time around THIS IS NOT ABOUT HIM; THIS IS NOT ABOUT POLITICS; THIS IS ABOUT SAVING LIVES. https://t.co/uy5dZY9uwY— Carmen Yulín Cruz (@CarmenYulinCruz) August 28, 2019
On Tuesday, Mr Trump tweeted: "Wow! Yet another big storm heading to Puerto Rico. Will it ever end?"
The president has previously faced political censure for his 2017 response to Hurricane Maria, which caused severe devastation in Puerto Rico.
He rated his handling of the disaster as a "tremendous success" while disputing official findings of the high death toll.
Has Puerto Rico recovered from Maria?
Some 30,000 homes in Puerto Rico still do not have proper roofs, merely tarps, according to US media.
The territory remains burdened with more than $70bn in debt - a crisis exacerbated by storms.
Nearly 3,000 of the island's over three million residents died as a result of Maria - many due to poor healthcare and a lack of electricity and clean water.
More than 1,000 roads remain blocked by that storm's landslides, the island's transportation secretary has said.
It took 11 months to restore full power to the island, and recurring electricity cuts caused further deaths from diabetes and sepsis.
The storm was the most intense cyclone worldwide that year and caused an estimated $100bn (£77bn) in damage.