Forecasters warned that Hurricane Otto may have winds of 90 mph when it makes landfall early on November 24.
At least four people have already died in Panama in severe weather caused by the approach of the storm.
Otto is moving west at 2 mph after being almost stationary throughout November 22, experts say.
According to the US National Hurricane Center, Otto is now blowing at about 75 mph as it approaches northern Costa Rica and southern Nicaragua.
The storm is now centered east of Limon, Costa Rica. When it makes landfall on November 24 its heavy rain is expected to create numerous mudslides. It is then expected to cross over into the Pacific.
The Washington Post says that very few hurricanes have formed so late in the season – which ends on 30 November – and that across the Atlantic since 1851 only 35 storms have reached tropical storm intensity on or after November 15.
Earlier in Panama two people died in a mudslide; a girl drowned in a river and a boy died when a tree fell on a car taking him to school. His mother, who was driving, survived.
According to officials, about 50 homes were destroyed by mudslides and flights were delayed.
The governments of Nicaragua and Costa Rica have issued a hurricane watch from Costa Rica’s southern border to the city of Bluefields in Nicaragua.
There is also a likelihood of dangerous surf and rip current conditions over the next few days along the coasts of Panama, Costa Rica and Nicaragua, officials say.
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