An image from a live video feed the oil plume is seen on the BP.com website early Wednesday, May 26, 2010. The oil company planned a "top kill" designed to choke off the gusher of oil at the bottom of the Gulf of Mexico by force-feeding it heavy drilling mud and cement early Wednesday May 26, 2010. (AP Photo/BP.com)
An operation using undersea robots to cut off the fractured pipe and seal it with a cap was launched on Tuesday.
"If everything goes well, within the next 24 hours, we could have this contained," Doug Suttles told the AFP news agency.
If the cap is fitted, the firm hopes to pipe most of the oil to the surface.
The bid follows the failure of the oil giant's "top kill" bid to stem the leak by pumping mud into the well. It could initially increase the flow and success is not guaranteed, BP has said.
On Sunday, White House energy adviser Carol Browner said the spill was the worst environmental disaster the US had ever faced.
She also said the US was "prepared for the worst scenario" that the leak might not be stopped before August.
Meanwhile, the National Oceanic and Atmospheric Administration warned that winds forecast later this week could move the spill towards the Mississippi and Alabama coasts, which have been less badly affected than Louisiana's shores.
Clean-up workers have been deployed on Louisiana beaches
The hurricane season was also due to begin on Tuesday, raising fears that high winds may whip the spill on to the nation's shores at a greater rate.
At least 20 million gallons (76 million litres) have now spilled into the Gulf, affecting more than 70 miles (110km) of Louisiana's coastline.
Reuters news agency reported that Attorney General Eric Holder is visiting the affected region on Tuesday to see the damage for the first time.
He was expected to meet legal officials from Louisiana, Alabama and Mississippi, prompting speculation over what future legal action may be faced by BP and other firms involved.
Eleven rig workers died when the Deepwater Horizon rig exploded and sank six weeks ago.
BP will attempt to lower its containment device - known as a Lower Marine Riser Package - over the damaged well.
As a first step, it will use undersea robots to slice through a damaged riser pipe to make a clean cut.
The containment cap would then be placed over the top and a new pipe will carry the captured oil to a ship on the surface.
However, BP has said the operation had never been carried out at a depth of 5,000ft (1,500m) and "the successful deployment of the containment system cannot be assured".
Mr Suttles said at the weekend that if it worked it would capture a majority of the spill, but would not stop it entirely.
The White House said the president had been informed that the flow rate could increase by as much as 20% until the containment device was applied over the leak.
The system is similar to a previous containment-dome plan that failed.
BP managing director Robert Dudley said the company would know by the end of the week if the latest attempt had succeeded.
Read more on News.BBC.co.uk:Gulf oil spill: BP hopes to cap flow 'within 24 hours'