BP says it is encouraged by the first test data following its stoppage of the oil from its leaking Gulf of Mexico well.
President Obama gave a cautious welcome but added: "It is important we don't get ahead of ourselves."
Pressure within the well is steadily rising, a good sign, said Kent Wells, BP's vice president.
The oil has been stopped for the first time since 20 April, as part of a 48-hour test.
There is currently no evidence of the well rupturing.
Eleven workers were killed in the Deepwater Horizon explosion, and the oil spill has raised fears of an environmental catastrophe.
The flow of oil was shut off at 1425 local time (1925 GMT) on Thursday. The stoppage is part of a test of the integrity of the well.
If the pressure within the new cap on the well stays high, that could mean there are no other leaks or ruptures within the wellbore. If it drops, that could suggest problems.
"The new cap is good news," Mr Obama said, noting that it would either mean the oil was stopped or that almost all of it would be able to be captured.
But he added: "One of the problems with having this camera down there is, is that when the oil stops gushing, everybody feels like we're done - and we're not."
The pressure within the well is currently 6,700 pounds per square inch (psi) and steadily rising, said Mr Wells.
If it were to drop below 6,000psi that would probably mean there was a problem within the well. If it continues rising and stays over 8,000psi that would probably mean the well was intact, Mr Wells said.
There is currently "no negative evidence of any breaching" of the sea floor, Mr Wells said. BP will soon run another seismic survey to check for any evidence of ruptures.
BP is also resuming work on a relief well that has 30ft left to drill before it hits the original leaking well. Once the wells intersect, mud and cement will be used to permanently deal with the leak.
The current pressure test could last for up to 48 hours, with BP and government experts reviewing results every six hours.
If the test is successful it is not clear what will happen next.
BP has suggested it might be possible to keep the well shut, with oil collection vessels left on standby.
But government incident commander Thad Allen has suggested the most likely outcome is the resumption of collection of the oil with four vessels and the capacity to collect 80,000 barrels - all or virtually all of the oil - each day.