The Atlantic hurricane season's first named storm posed an uncertain threat to the Gulf of Mexico on Sunday, as a report said relief wells may halt the worst U.S. oil spill ahead of schedule.

Alex, downgraded to a tropical depression but expected to regain storm strength over the Gulf of Mexico later on Sunday, does not pose an imminent threat to oil-siphoning efforts at BP Plc's blown-out well off the Louisiana coast.

But even a miss that generates large waves could greatly complicate clean-up efforts from Louisiana to Florida from the undersea leak that has gushed oil since April 20, threatening fisheries, tourism and wildlife.

Alex has at least a moderate chance of becoming a major hurricane as it passes over the Gulf and makes an expected landfall later in the week between Brownsville, Texas, and Tuxpan de Rodriguez Cano in Mexico, the U.S. National Hurricane Center said.

RELIEF WELLS

Official estimates suggest between 35,000 and 60,000 barrels of oil per day are spewing from BP's damaged well on the seabed about a mile below the surface.

The energy giant estimated its siphoning efforts collected more than 24,000 barrels on Friday and about 11,640 barrels in the first half of Saturday.

Britain's Sunday Times newspaper said BP could plug the leak in mid-July, two weeks before its guidance of early August. The drilling of two relief wells is going faster than expected, it said, citing sources familiar with the operation.

A BP spokesman declined to comment on the report and referred to a company statement on Friday that the relief wells would take about three months to complete. Drilling of the wells began in May.

Equipment going to the leak site this week could raise daily collection to 53,000 barrels, officials say, and a review is scheduled of a system that may boost it to 80,000 barrels.

But progress on relief wells and other efforts could halt if Alex or a subsequent storm comes too close.

Sandy Palmer sells pickled quail eggs, raspberry jam, kumquat trees and dozens of other odds and ends at her farmers market on the side of Louisiana's Rt. 23 in Belle Chasse.

While Alex isn't currently expected to strike anywhere near her, she's still nervous. Even if we just have the threat of a storm we have to take everything down, said Palmer.

SUMMIT TOPIC

The Gulf disaster and its impact on UK-based BP were on the agenda on Saturday when President Barack Obama and British Prime Minister David Cameron met at a summit in Canada.

Shares of BP, a staple holding of many British pension funds, have been savaged since the crisis started and fell another 6 percent to a 14-year low on Friday.

Investors fret about the costs to BP, which include but are not limited to a $20 billion compensation fund it set up under intense U.S. political pressure. BP said it has paid out $2.35 billion so far in clean-up and compensation costs.

Obama has been highly critical of BP, even as his own poll ratings have fallen amid perceptions his handling of the Gulf crisis has been too slow.

Cameron and Obama agreed that there was nothing to be gained from damaging BP as a going concern, a British official said after the leaders met at the G8/G20 summit. They also agreed BP must meet its obligations to cap the leak, clean up the damage and pay legitimate compensation.

The compensation process has met controversy over its fairness and paperwork requirements.

Residents of southeast Louisiana fishing areas said on the weekend they saw some improvements, but bureaucracy still reigned supreme.

Shrimp boat captain Louis Bateaux, 69, only got half what captains normally receive on his first visit to a claims center, then made a three-hour drive to get more evidence of his skipper status.

Fishermen should be first in line. We need the money, he said.

Kenneth Feinberg, an attorney and Obama's newly appointed oil fund czar, has promised to streamline the process and even let some victims file claims online.

LEGAL WRANGLING, PROTESTS

Many in the region also complain about a six-month ban on new deepwater drilling in the Gulf. Farmers market operator Palmer said her business has dropped sharply since the oil leak and deepwater drilling moratorium began.

Oil workers used to stop by when their shifts ended. Since the moratorium, my business has gone way down.

A federal judge overturned the ban, but the Obama administration asked a U.S. appeals court to stay the ruling.

On Saturday, the state of Louisiana filed a legal brief opposing the administration request. Each day the ban is in place, millions of dollars of income are lost to the citizens of Louisiana, and by the state, the brief said.

The disaster is taking a heavy toll on fishing, tourism and the environment. About a third of U.S. federal waters in the Gulf are closed to fishing.

The U.S. Fish and Wildlife Service reports a growing toll of birds, sea turtles and marine mammals, mostly dolphins, found dead or debilitated along the Gulf Coast.

In one of the latest cases against BP, Susan Spicer, among New Orleans' most prominent and highly regarded chefs, sued the company for damages to restaurants that have lost normal seafood supplies because of the spill.

More than 250 lawsuits have been filed over alleged damage from the incident, according to the Westlaw database, a Thomson Reuters unit.

(Additional reporting by Ernest Scheyder in Grande Isle, Sarah Young in London, Jonathan Stempel in New York, Sumeet Desai in Toronto, Jose Cortazar in Cancun, Jeremy Pelofsky in Washington, and Mike Peltier in Florida; Writing by Jerry Norton; Editing by John O'Callaghan and Vicki Allen)