The effort would involve pumping drilling mud through the top of the cap and into the wellbore to stop up the oil flow. The idea is similar to the failed top kill plan that couldn't overcome the pressure of the geyser pushing up.
BP said it could work now because there's less oil to fight against, but it wasn't clear how such a method would affect the cap's stability.
A decision on static kill is expected in the next few days, reports CBS News correspondent Kelly Cobiella.
However, retired Coast Guard Adm. Thad Allen said the relief wells currently being drilled were still considered the plan for a permanent fix.
Meanwhile, federal officials remain unfazed by oil seeping out of BP's blown out well. The oil is seeping more slowly and scientists aren't convinced the cap that stopped the flow last week is making things worse.
The government said Monday that oil was seeping into the Gulf after days of warning that the experimental cap on the oil well could cause more leaks.
Despite what at first seemed a setback, though, the federal government declared the development insignificant and forged ahead with BP's plan for finally sealing the hole in the ocean floor.
Ever since the cap was used to bottle up the oil last week, engineers have been watching underwater cameras and monitoring pressure and seismic readings to see whether the well would hold or spring a new leak, perhaps one that could rupture the sea floor and make the disaster even worse.
Small amounts of oil and gas started coming from the cap late Sunday, but "we do not believe it is consequential at this time," Allen said.
Also, seepage from the sea floor was detected over the weekend less than two miles away, but Allen said it probably has nothing to do with the well. Oil and gas are known to ooze naturally from fissures in the bottom of the Gulf of Mexico.
At a Monday afternoon briefing in Washington, Allen said BP could keep the cap closed at least another 24 hours, as long as the company remained alert for leaks.
Since the cap was closed Thursday, beachgoers have reported less oil fouling the shore.
Bob Broadway, 41, of Huntsville, Ala., said his vacation spot in Orange Beach, Ala., has improved from a month ago.
Then, he said, the oil was thick "like chocolate" and the beach smelled like "an old mechanic's garage."
"The beach looks better now than before," he said Monday.
BP and the government had been at odds over the company's desire to simply leave the cap in place and employ it like a giant cork in a bottle until a relief well being drilled deep underground can be used to plug up the well permanently.
Allen initially said his preference was to pipe oil through the cap to tankers on the surface to reduce the slight chance that the buildup of pressure inside the well would cause a new blowout. That plan would require releasing millions more gallons of oil into the ocean for a few days during the transition a spectacle BP apparently wants to avoid.
On Monday, Allen budged a bit, saying unless larger problems develop, he's not inclined to open the cap.
Work on a permanent plug is moving steadily, with crews drilling into the side of the ruptured well from deep underground. By next week, they could start blasting in mud and cement to block off the well for good. Killing the well deep underground works more reliably than bottling it up with a cap.
Somewhere between 94 million and 184 million gallons have gushed into the Gulf over the past three months in one of America's worst environmental crises.
BP PLC said the cost of dealing with the spill has now reached nearly $4 billion. The company said it has made payments totaling $207 million to settle claims for damages. Almost 116,000 claims have been submitted and more than 67,500 payments have been made. BP stock was down slightly Monday.