A state of emergency has been declared in the Central American countries, where more than 20 people are missing.
Nate has caused heavy rains, landslides and floods which are blocking roads, destroying bridges and damaging houses.
In Costa Rica, nearly 400,000 people are without running water and thousands are sleeping in shelters.
At least 8 people have died in the storm there, while another 11 were killed when it moved north and reached Nicaragua, where as much as 15ins of rain had been predicted to fall by the US’s National Hurricane Center.
Three people have been killed in Honduras, including two youths who drowned in a river, and several are reported missing.
One man was also reportedly killed in a mudslide in El Salvador, according to emergency services.
On October 5, all train journeys were suspended and dozens of flights canceled in Costa Rica.
More than a dozen national parks popular with tourists have been closed as a precaution.
Nate also caused extensive damage to infrastructure in Nicaragua.
Nicaragua’s VP Rosario Murillo advised people to be cautious in the heavy rains.
“Sometimes we think we think we can cross a river and the hardest thing to understand is that we must wait,” he said on state radio.
“It’s better to be late than not to get there at all.”
According to the National Hurricane Center, the storm was expected to move off the eastern coast of Honduras at 23:00 ET on October 5.
Forecasters say Nate will gain strength and become a category one hurricane before it makes landfall on the southern coast of the United States on October 8.
Residents from Florida to Texas have been told to prepare for the storm, which, if it does strike, might be the third major storm to hit the southern coast his year.
Both Texas and Florida are still recovering from the damage inflicted by Hurricane Harvey, which hit the former in August and caused “unprecedented damage”, and Hurricane Irma, which made landfall in Florida in September.
A state of emergency was declared in 29 Florida counties, and in New Orleans in Louisiana.
New Orleans Mayor Mitch Landrieu told people who live on low-lying ground to evacuate.
He tweeted: “There is no need to panic.
“Be ready and prepare. Get a plan. Prepare to protect your personal property.”
Oil companies operating in the Gulf of Mexico says they are evacuating staff from platforms which lie along the predicted path of Tropical Storm Nate.
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