But under federal law -- the War Powers Resolution of 1973 -- Obama is only allowed to keep U.S. forces engaged in hostilities for 60 days, unless Congress declares war, authorizes funding for the effort or extends the deadline.
Congress has not enacted legislation authorizing military involvement in Libya, and the White House has not made a public effort to comply with the rule.
Experts say this is the first time an American president has defied the War Powers Resolution's deadline for participation in combat operations without any concurrent steps by Congress to fund or otherwise authorize the role.
While every administration since 1973 and some lawmakers have questioned the constitutionality of the resolution's infringement on executive power, it has not been successfully challenged in court and remains the law of the land.
Now some members of Congress from both parties want answers.
"As recently as last week your administration indicated use of the U.S. armed forces will continue indefinitely. Therefore, we are writing to ask whether you intend to comply with the requirements of the War Powers Resolution," six Republican senators wrote Wednesday in a letter to Obama. "We await your response."
Sen. Richard Lugar of Indiana, the ranking member of the Senate Foreign Relations Committee, did not co-sign the letter but has been pressing the administration to seek Congressional authorization on Libya for weeks.
"Extended engagements in military action abroad, and the costs and risks they entail, must be undertaken only with the full support of the American people," Lugar said last week.
U.S. involvement in the Libya operation is estimated to have cost taxpayers at least $750 million so far, according to the Pentagon. U.S. drones and military aircraft, including refueling tankers and surveillance planes, remain actively involved in enforcing a no-fly zone above Libya.
White House press secretary Jay Carney has said that the administration plans to continue "consultations" with Congress on Libya but would not comment when asked recently about the passing of the 60-day deadline.
But Deputy Secretary of State James Steinberg told a Senate panel last week that Obama remains "mindful" of the War Powers Resolution and seeks to abide by it.
"He will continue to do so, and we look forward to continuing to consult with Congress on our role in the coming days," he said.
With the House of Representatives on recess and Obama headed to Europe next week, many observers don't see a speedy resolution to the issue and expressed disappointment at the impasse.
"They're making a mistake on the War Powers Act," said Brookings Institution foreign policy fellow Michael O'Hanlon. "Congress does deserve a role in this and making the case to Congress for continued military action in Libya shouldn't be hard."
Tom Donnelly, director of Center for Defense Studies at the right-leaning American Enterprise Institute, speculated that Obama has tried to avoid the issue of extending operations in Libya given public sentiment on the conflict.
"There's been rising domestic and political discontent about it," he said. "Maybe he [Obama] can get away with it because of the Osama bin Laden killing. But it's difficult to say in the games of chicken between the White House, Capitol Hill and all over town."
"This is a war entirely at our disposal to bring to a conclusion given our military power. If we really cared about it, it could have been over and done with by now," Donnelly said.
Recent polls show a strong majority of Americans approve of Obama's handling of national security issues and foreign affairs. But on Libya he has not had significant support from either side of the political spectrum.
Forty percent of Americans oppose U.S. military involvement in Libya, and among those, 65 percent disapprove of his handling of the situation so far, according to an ABC News/Washington Post poll conducted last month. Some conservatives think Obama should have been more aggressive in using force to oust Gadhafi.
For his part, Obama has sought to take a backseat in the Libyan intervention, stressing the U.S. holds a support role as part of an international coalition led by NATO.
"We cannot prevent every injustice perpetrated by a regime against its people, and we have learned from our experience in Iraq just how costly and difficult it is to impose regime change by force -- no matter how well-intended it may be," Obama said Thursday in a speech on Middle East policy.
"But in Libya, we saw the prospect of imminent massacre, had a mandate for action, and heard the Libyan people's call for help. Had we not acted along with our NATO allies and regional coalition partners, thousands would have been killed."
Still, nearly four years ago, it was then-candidate Obama who suggested an end didn't justify the use of any means.
"No more ignoring the law when it's inconvenient. That is not who we are," Obama said, referring to his predecessor President George W. Bush. "We will again set an example for the world that the law is not subject to the whims of stubborn rulers."