Mr Obama reportedly signed the order, known as a presidential "finding", within the last two or three weeks, four US government sources told Reuters.
Such findings are a principal form of presidential directive used to authorise secret operations by the Central Intelligence Agency. The CIA and the White House declined immediate comment.
News that Mr Obama had given the authorisation surfaced as the President and other US and allied officials spoke openly about the possibility of sending arms supplies to Gaddafi's opponents, who are fighting better-equipped government forces.
In interviews with American TV networks on Tuesday, Mr Obama said the objective was for Gaddafi to "ultimately step down" from power. He spoke of applying "steady pressure, not only militarily but also through these other means" to force Gaddafi out.
Mr Obama said the US had not ruled out providing military hardware to rebels. "It's fair to say that if we wanted to get weapons into Libya, we probably could. We're looking at all our options at this point," the President said.
US officials monitoring events in Libya say that at present, neither Gaddafi's forces nor the rebels, who have asked the West for heavy weapons, appear able to make decisive gains.
While US and allied air strikes have seriously damaged Gaddafi's military forces and disrupted his chain of command, officials say, rebel forces remain disorganised and unable to take full advantage of western military support.
People familiar with US intelligence procedures said that Presidential covert action "findings" are normally crafted to provide broad authorisation for a range of potential US government actions to support a particular covert objective.
In order for specific operations to be carried out under the provisions of such a broad authorisation – for example the delivery of cash or weapons to anti-Gaddafi forces – the White House also would have to give additional "permission" allowing such activities to proceed.
Former officials say these follow-up authorisations are known in the intelligence world as "'Mother may I' findings."
In 2009 Mr Obama gave a similar authorisation for the expansion of covert US counter-terrorism actions by the CIA in Yemen. The White House does not normally confirm such orders have been issued.
Because US and allied intelligence agencies still have many questions about the identities and leadership of anti-Gaddafi forces, any covert US activities are likely to proceed cautiously until more information about the rebels can be collected and analysed, officials said.
According to an article speculating on possible US covert actions in Libya published early in March on the website of the Voice of America, the US government's broadcasting service, a covert action is "any US government effort to change the economic, military, or political situation overseas in a hidden way."
The article, by VOA intelligence correspondent Gary Thomas, said covert action "can encompass many things, including propaganda, covert funding, electoral manipulation, arming and training insurgents, and even encouraging a coup."
Members of Congress have expressed anxiety about U.S. government activities in Libya. Some have recalled that weapons provided by the US and Saudis to mujahideen fighting Soviet occupation forces in Afghanistan in the 1980s later ended up in the hands of anti-American militants.
There are fears that the same thing could happen in Libya unless the US is sure who it is dealing with. The chairman of the House intelligence committee, Rep. Mike Rogers, said on Wednesday he opposed supplying arms to the Libyan rebels fighting Gaddafi "at this time."
"We need to understand more about the opposition before I would support passing out guns and advanced weapons to them," Mr Rogers said in a statement.