The U.S. National Hurricane Center in Miami said Cristobal may strengthen into a hurricane later in the week over the open Atlantic. The storm's center was expected to curve away from the U.S. East Coast.
Many residents in the sparsely populated southeastern Bahamas and the tiny British Caribbean dependency of the Turks and Caicos Islands hunkered down as Cristobal's rains pelted windowpanes.
Capt. Stephen Russell, head of the Bahamas' emergency management agency, said there had been no reports of damage. Air traffic to the southeastern Bahamian islands had not been suspended, but sea vessels were advised to remain in port, he said.
Turks and Caicos Premier Rufus Ewing advised residents to remain indoors as much as possible because the island chain south of the Bahamas was still experiencing heavy rains and "extensive flooding in low-lying areas," especially on Middle Caicos and North Caicos islands.
"The inclement weather is expected to linger for another 48 hours and the flooding is expected to worsen as a result," Ewing said in a Sunday statement.
Cristobal, which formed as a tropical depression over the Turks and Caicos Islands on Saturday, is the fourth depression of the Atlantic hurricane season.
By late Sunday afternoon, the tropical storm had sustained winds near 45 mph (75 kph) and was located about 155 miles (250 kilometers) east-northeast of the Bahamas' Long Island.
The rain-heavy storm was tracking north at about 8 mph (13 kph). U.S. forecasters said there should be a decrease in forward speed over the next couple of days, meaning Cristobal's center is expected to move near to or east of the central Bahamas through Monday.
U.S. forecasters say lingering Cristobal could bring up to 8 inches (20 centimeters) of rain to the Turks & Caicos Islands and southeastern and central Bahamas through Tuesday.
Before strengthening into a storm, it had downed several trees and power lines on Puerto Rico, leaving more than 23,500 people without power and 8,720 without water. There were a handful of reported landslides.
Police said in a statement that a small bridge collapsed Saturday in the central Puerto Rican town of Barranquitas, isolating some 25 families in the area. No one was injured.
AP writer David McFadden reported from Kingston, Jamaica, and Evens Sanon contributed from Port-au-Prince, Haiti.