Work to install a new, tighter containment cap on the well head will begin Saturday, Allen said at a New Orleans news conference. At the same time, an additional ship to collect escaping oil, the Helix Producer I, is expected to come online Sunday.
The new cap, nicknamed Top Hat No. 10, is currently on the deck of the Discoverer Enterprise, one of the ships on the surface sucking up oil.
"We're going to attempt to remove the current cap [Saturday]," Allen said. "That will start the sequence. At the same time, we will bring the Helix Producer into production. We're going to have to take Discoverer, connected to the riser, and move it to the side."
Three vessels will be in place to capture oil from three different lines coming up from the well.
The Discoverer Enterprise is hooked up directly to the riser -- the main pipe coming from the well -- and collects 15-18,000 barrels a day. Another ship, the Q400, is connected to a smaller "choke line" and produces 5-10,000 barrels a day. The Helix Producer will connect to a third line -- known as the "kill line" -- and produce 20-25,000 barrels.
If all goes according to plan, the new cap and the ships could start to contain all the leaking oil by Monday, stopping it from escaping into the Gulf for the first time since the Deepwater Horizon drilling platform caught fire and the disaster began April 20.
"I use the word 'contained,'" Allen said. "'Stop' is when we put the plug in down below."
Still, Allen offered cautious hope that the blowout might finally come under control in the next few days.
"This is in anticipation of a weather window that will allow us seven to 10 days of good weather," Allen said. "We'd like to take advantage of that."
Depending on whose estimates one believes, the BP disaster may now be the biggest accidental oil spill in history, exceeding the 140 million gallons of oil estimated to have escaped into the Gulf of Mexico in 1979 and 1980 from the Mexican-owned Ixtoc 1 well 800 miles to the south.
Gulf of Mexico Spill: Well Could be Under Control in Days, Says Commander
Though Allen said Monday was a best-case scenario, he said it was likely that by the end of next week, pressure in the new cap could be tested to determine if it was stable enough to kill the leak.
The valves in the new cap will be shut. If pressure reads 9,000 pounds per square inch at the cap, it is likely the pipe can be filled with mud and cement in order to seal it.
If this cap doesn't work, the producing vessels at the surface will continue to suck up oil from three different underwater pipes until another cap can be tried.
In the meantime, drilling continues on two relief wells, which BP said would cut off flow from the original blown-out well by mid-August.
As of this morning, the first relief well was at 17,700 feet measured depth below the surface. Engineers said it can intercept oil from the original well at 17,874 ft.
"We're close," Allen said.
But despite closing the distance, he is still being conservative about the timetable.
He said he is still predicting a mid-August date for making contact "cutting the casing and filling it with cement."