Obama's order, approved earlier this year and known as an intelligence "finding", broadly permits the CIA and other US agencies to provide aid that could help the rebels dislodge Assad from power.
The shift towards supporting Assad's armed opponents intensified following last month's failure of the UN Security Council to agree on tougher sanctions against the Damascus government.
The full extent of clandestine support that agencies like the CIA might be providing also is unclear. White House spokesman Tommy Vietor declined comment on the matter.
The White House has apparently stopped short of giving the rebels lethal weapons, even as some US allies have been doing just that
US and European officials have said that there have been noticeable improvements in the coherence and effectiveness of Syrian rebel groups in the past few weeks. That represents a significant change in assessments of the rebels by Western officials, who previously characterised Assad's opponents as a disorganised.
Separately from the president's secret order, the Obama administration has stated publicly that it is providing some backing for Assad's opponents.
The State Department said on Wednesday the US government had set aside a total of $25m for "non-lethal" assistance to the rebels.
A US official said that was mostly for communications equipment, including encrypted radios.
The State Department also says the US has set aside $64m in humanitarian assistance for the Syrian people, including contributions to the World Food Program, the International Committee of the Red Cross and other aid agencies.
A US government source acknowledged that under provisions of the presidential finding, the US was collaborating with a secret command centre operated by Turkey and its allies.
Last week, Reuters reported that, along with Saudi Arabia and Qatar, Turkey had established a secret base near the Syrian border to help direct vital military and communications support to Assad's opponents.
This "nerve centre" is in Adana, a city in southern Turkey about 100km from the Syrian border, which is also home to Incirlik, a US airbase where US military and intelligence agencies maintain a substantial presence.
Turkish authorities are said to be increasingly involved in providing Syrian rebels with training and possibly equipment.
European government sources said wealthy families in Saudi Arabia and Qatar were providing significant financing to the rebels. Senior officials of the Saudi and Qatari governments have publicly called for Assad's departure.
On Tuesday, reports emerged that the Free Syrian Army had obtained nearly two dozen surface-to-air missiles, weapons that could be used against Assad's helicopters and fixed-wing aircraft.
Syrian government armed forces have employed such air power more extensively in recent days.
NBC network said the shoulder-fired missiles, also known as MANPADs, had been delivered to the rebels via Turkey.
On Wednesday, however, Bassam al-Dada, a political adviser to the Free Syrian Army, denied the NBC report, telling the Arabic-language TV network Al-Arabiya that the group had "not obtained any such weapons at all".
US government sources said they could not confirm the MANPADs deliveries, but could not rule them out either.
Current and former US and European officials previously said that weapons supplies, which were being organised and financed by Qatar and Saudi Arabia, were largely limited to guns and a limited number of anti-tank weapons, such as bazookas.