US National Hurricane Center traced storm centre on Saturday afternoon to 45 miles (75km) south-west of Morgan City, with maximum winds of 60mph (95km/h).
Flood defences repaired after the 2005 disaster are expected to be put to the test in New Orleans.
According to National Hurricane Center forecasters, Tropical Storm Lee had been stationary for a few hours then began a slow and erratic path to the north and north-east, and is expected to make landfall later on Sunday.
Lee comes few days after Hurricane Irene killed 44 people from North Carolina to Maine and left millions of people without electricity.
For the moment is too soon to tell if another hurricane, Katia, which is out in the Atlantic, could threaten the US.
The Atlantic hurricane season usually brings about a dozen named storms, but Katia is already the 11th with half the season still ahead.
New Orleans Mayor Mitch Landrieu has warned of possible major flooding, with up to 10in (25cm) of rain forecast for the city.
The Army Corps of Engineers, which operates the flood control structures in New Orleans, says it is not planning to close any of the structures yet.
“People are probably scared to death to come here after Katrina,” Ann Sonnier, shift manager of Jester’s bar, told the Associated Press news agency, adding that receipts had been disappointing so far.
Nearly half of US Gulf oil production was shut on Friday as companies – including Exxon Mobil, Shell and Chevron – evacuated workers and shut offshore platforms.
BP, the largest oil producer in the Gulf of Mexico, told the BBC it had evacuated all personnel and shut down production.
Louisiana Governor Bobby Jindal warned residents to “make sure they have a game plan for themselves and their families should this storm strengthen”.
However, the developing weather system could bring some much-needed rain to Texas, which is in the grip of a severe drought.
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