The 98-tonne, dome-topped box being built by steel workers will be connected to a drill ship and lowered to the sea floor.
BP has never deployed such a structure at a depth of 5,000ft (1,500 metres) and difficulties may occur, it says.
"There's no guarantees," BP boss Tony Hayward told AP news agency.
"We'll undoubtedly encounter some issues as we go through that process.
"But if that was a good outcome, then you would have the principal leak contained by the early part of next week," Mr Hayward said.
BP has accepted it is "absolutely responsible" for cleaning up the spill caused by a blast on a drilling rig.
Its shares have plunged more than 15% since the 22 April explosion.
It is estimated that the clean-up could cost BP up to $15bn (£10bn).
BP insists that the rig's contractor, Transocean, was to blame for the accident but, under US law, it has to bear the cost of the clean-up as the operator of the oilfield.
Both companies are expecting lawsuits over the slick, which US President Barack Obama described as a "potentially unprecedented" environmental disaster.
Californian Governor Arnold Schwarzenegger has withdrawn his support for a plan to expand oil exploration off that state's coast, saying he has changed his mind about the safety of oil platforms in the Pacific Ocean.
After seeing television pictures of the Gulf of Mexico spill, he asked: "Why would we want to take that kind of risk?"
Technical challenges
The 40ft (12m) funnel resembles a primitive space rocket with a hole on top to channel oil through a pipe from the sea floor to the surface where it can be collected on a barge, Reuters news agency reports.
It is being built at Golden Meadow, near Port Fourchon in southern Louisiana, by Wild Well Control, a company that specialises in controlling oil fires.
A team of 25 people is working around the clock on the funnel, marine engineer Jason Holvey told the agency.
In theory, the system should collect 85% of the oil rising from the sea floor.
While the precise location of the leak is known, work will have to be carried out in the face of severe technical challenges, not least because of the softness of the sea floor.
Asked by Reuters if he was confident it would work, BP spokesman John Curry said: "We sure hope so.
"If not, we will keep trying other options until something does work. I think we have some great engineers and they have worked very hard."
The funnel will have to be tested before it can be shipped to the spill site.
A second, back-up funnel is also under construction, AP said.
Mr Holvey said similar containment devices had been used in the Gulf before, but in shallow waters. They were used, for instance, after Hurricane Katrina to channel oil to the surface that had spilled from platforms.
Pressure on BP
The US government has been putting pressure on BP to act quickly.
In a BBC interview, Hayward dismissed talk of a rift between BP and US officials, saying an "incredible co-operative relationship" had been established with the federal authorities.
Thousands of barrels of oil have been leaking into the Gulf of Mexico every day since the rig sank.
Some oil has washed ashore, but officials say the bulk of the slick remains a few miles from the Louisiana coastline.
The oil is already having a devastating effect on the area's fishing industry, and officials fear much wider environmental damage if the full slick hits land.
Other efforts being made by BP, according to Mr Hayward, include using undersea robots to work on the safety valve which failed, allowing the oil to escape, and drilling a new well to help relieve the pressure and stem the flow from the rupture.
BP has said it will honour legitimate claims for compensation from people affected.
