(CNN) -- BP moved closer on Monday to sealing, once and for all, its ruptured well in the Gulf of Mexico, though government and company officials said their struggle against the spill's extensive environmental and economic damage is far from over.
A relief well is expected to start the final shutdown within the week, said retired Coast Guard Adm. Thad Allen, the government's point man in the region.
Allen said the closer of two relief wells alongside the capped, formerly gushing BP well in the Gulf is 17,909 feet deep and less than 100 feet from intercepting the main well. During the previous 72 to 96 hours, he said, crews had twice drilled for 30 feet at a time, then backed out and put wire down the pipe to gauge the exact location relative to the main well.
Allen said drillers would make "one more run," and hoped to intercept, then pump cement into, the main well as early as the end of the week, permanently sealing the crippled well.
Crews will have to keep a close watch on inclement weather while drilling. Strong thunderstorms and gusty winds are possible over the main well starting on Wednesday, said CNN meteorologist Sean Morris.
The National Hurricane Center said a broad area of low pressure over southern Florida is forecast to move slowly northwestward into the Gulf and toward the north-central Gulf Coast this week. The system has roughly a 30 percent chance of developing into a tropical cyclone in the next 48 hours, the center said.
Also on Monday, BP, the oil company that operated the sunken well, released an updated tally on expenses associated with cleanup and containment, which now stands at about $6.1 billion.
BP said it has established a trust and made an initial $3 billion deposit of the previously announced $20 billion escrow account to pay "legitimate" claims arising from the April 20 Deepwater Horizon incident and the resulting oil and gas spill.
The oil giant named two trustees who will administer the account: John S. Martin, a former federal judge for the Southern District of New York, and Kent Syverud, dean of the Washington University School of Law. Citigroup will serve as corporate trustee and paying agent for the account, BP said.
Responding to the company's $3 billion deposit, the Justice Department said Monday details of the account have been worked out. A top government attorney noted that BP is fulfilling its obligations.
"We are pleased that BP made an initial contribution and has taken an important step toward honoring its commitment to the president and the residents and business owners in the Gulf region," Associate Attorney General Tom Perrelli said in a statement confirming BP's $3 billion installment.
Also on Monday, BP's newly-appointed Chief Executive Bob Dudley met with high-level White House officials and Allen to talk about recovery efforts.
Government officials stressed the importance of BP living up to its commitments, including the quick payment of claims made by state and local governments, according to a White House statement.
President Barack Obama, who has been under pressure to show a strong response to the spill, said he was committed to standing by communities along the Gulf Coast well beyond when the well is closed.
"What is clear is that the battle to stop the oil flowing from into the Gulf is just about over. Our work goes on, though," he said in Washington.
The oil spill hasn't just hurt BP's bottom line -- it's inflicted heavy blows on Gulf coast industries like tourism and fishing.
A group of concerned St. Louis, Missouri, residents said they left on a caravan of support on Monday, spending money at small businesses along the Gulf Coast, using funds raised from around the country. The caravan is expected to travel through Mississippi, Alabama and the Florida panhandle, completing its trek on Friday.
"We've learned of so many businesses in the Gulf region that are losing their customers, employees and dreams because of the impact on tourism," organizer Dennis Gorg said in a statement. "As a small business owner, I can't imagine how I'd support the people who depend on me. We can do something. We can become tourists with a purpose."
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 oil back into the ocean floor last week.
The well gushed an estimated 53,000 barrels (2.3 million gallons) of oil per day before it was capped, with some of the oil ending up on beaches or in marshes. 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.
Allen said Monday responders will be "redoubling" their efforts in the marshes.
"There's a lot of clean-up working going on here, and that's excellent," Maura Wood of the National Wildlife Federation, told CNN last week. "But the tide keeps coming and going each and every day."
CNN's Terry Frieden, Eric Fiegel, Ed Lavandera and David Mattingly contributed to this report.