The company is also trying to drill a relief well parallel to the leaking one, but that could take three months.
The fastest fix would be to place rectangular steel boxes, that weigh 74 tons, over the leaking pipe and then funnel the oil up to a ship. The boxes are expected to be on location next weekend.
Napolitano said BP has seen some success in using underwater dispersants and will conduct a review of the method this morning.
Napolitano and Secretary of the Interior Ken Salazar will meet with BP executives in Washington, D.C., today.
Napolitano said she wants to ensure that people who have been affected by the oil slick have a clear claims process and "prompt reimbursement."
"They are going to pay for the federal government's cost, for the states' and most importantly for the individuals and communities that are going to be most directly impacted," Napolitano said.
The spill threatens four Gulf states and hundreds of miles of sensitive coast line.
So far, crews have found relatively few animals affected by the spill. An oil-soaked bird was discovered in Louisiana last week.
There were 25 dead sea turtles found on the shores of Mississippi in the last three days, though their deaths may not have been related to the spill. Necropsies conducted on some of the turtle bodies suggested that oil did not cause the deaths, though veterinarians didn't rule out the possibility.
More than 6,800 square miles of federal fishing areas, from the Mississippi River to the Florida Panhandle, have been closed for at least 10 days.
In parts of Alabama, the Army National Guard is trying to save the beach by create a high-tech barrier. Boxes filled with chemicals that soldify oil are being strung in the surf, so that when the oil makes contact it can be easily collected.
In other coastal areas, work crews are still placing booms, hoping they'll be enough to keep the oil offshore.
BP's safety record has been called into question, including a 2005 explosion at a Texas City refinery. The Wall Street Journal quoted Jordan Barab, a deputy assistant secretary of labor at the Occupational Safety and Health Administration, who said, "there is a systemic safety problem across the company."
Hayward said the company has improved its safety record.
"I think we have made enormous strides as a company in the last three-four years with a remorseless focus on safe and reliable operations," Hayward said.
BP is now concentrating on stopping the leak and cleaning the spill, he said.
"We are clearly focused on minimizing the overall impact," Hayward said. "We are a big company and we intend to deal with this. We take this responsibility incredibly seriously. We absolutely will prevail and we will deal with it."
The Associated Press and ABC News' Jeffrey Koffman, Ryan Owens, Ayana Harry and Sunlen Miller contributed to this report.