Earlier, officials had said the storm would stall the crucial work for about four days.
As is currently stands, the Development Driller III, the rig that is drilling the relief well, is cleaning the area out ahead of drilling the remaining 30 to 50 feet to reach the Macondo well, BP spokesman Robert Wine said.
Officials said they were looking at announcing a permanent kill, weather permitting, by the end of next week.
The weather system could bring 3 to 5 inches of rain to the region through Friday morning, with isolated pockets of up to 8 inches, the National Hurricane Center said.
The well erupted after an April 20 explosion aboard the drilling rig Deepwater Horizon that left 11 men dead. A temporary cap contained the spill on July 15, and nearly 3,000 barrels of heavy drilling mud and cement drove the well back into the ocean floor last week through a process called a "static kill."
The well gushed an estimated 53,000 barrels (nearly 2.3 million gallons) of oil per day before it was capped. Since then, fresh, green grass has begun growing again in some of the hardest-hit marshes of southern Louisiana, but oil continues to wash ashore in places.
Some of the environmental damage appears to have abated. The National Oceanic and Atmospheric Administration reopened 5,144 square miles of Gulf waters on Tuesday for commercial and recreational fishers to catch finfish, saying that since July 3, its data have shown no oil in the area. Coast Guard observers flying over the area in the past 30 days also have not observed any oil. In addition, NOAA said fish caught in the area and tested by NOAA experts have shown no signs of contamination.
The closed area now covers 52,395 miles, or 22 percent of the federal waters in the Gulf, down from 37 percent at its peak, NOAA said.
The Gulf remains closed to deepwater drilling, but the head of the government agency that regulates offshore drilling said in Mobile, Alabama, on Tuesday it's unlikely a six-month moratorium on the practice will be extended.
"Obviously, we can't predict everything that we learn or everything that may happen in the outside world before then, but ... I see no information so far that would justify extending the moratorium," said Michael Bromwich, the head of the Bureau of Ocean Energy Management, formerly the Minerals Management Service. "[It's] not impossible but unlikely," he said.
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