But there are reports that the Americans and their Western allies, fearful of the jihadists' growing power and the threat they could pose, have been urging the so-called moderate rebel groups to move against the Islamist militants linked to al-Qaida.
There has been repeated skirmishing during the 28-month-old conflict between the jihadists, led by Jabhat al-Nusra, or the al-Nusra Front, and the Islamic State of Iraq and Levant, and the more secular groups like the Free Syrian Army which is part of the pro-Western Supreme Military Council.
But the July 11 assassination of Kamal Hamami, a leader of the SMC, and his brother, which the FSA claims was carried out in northern Latakia province by the al-Qaida affiliated ISIL, sharply escalated the feuding and could cause immense damage to the fragmented rebel cause if it's allowed to continue.
The FSA demanded the assassins be handed over. The ISIL did not do so and heavy clashes were reported between the rival factions in the northern city of Aleppo.
This was not the first such episode. Both sides have killed each others' commanders fairly regularly, usually over competition for resources, as seemed to be the case with Hamami, rather than ideology.
But Hamami was notable for his seniority and for his close links with the West at a time when the "moderates" appear to be under U.S. pressure to move against the Islamists in force.
Hamami belonged to a faction led by Gen. Selim Idriss, overall leader of the SMC and the main link to the Americans and their European allies who have been working toward arming non-jihadist opposition groups already being supplied by Saudi Arabia and Qatar.
These killings left Idriss with the unquestionably difficult task of trying to curtail al-Qaida's influence on the rebellion without bringing about the collapse of the disparate, largely Sunni opposition movement seeking to bring down Assad's regime, dominated by Syria's Alawite minority.
Idriss "is in a very difficult position, as are those around him," observed Salman Sheikh, director of the Brookings Doha Center think tank in Qatar. "And the regime will benefit more than anyone else."
The al-Nusra Front and the ISIL bitterly oppose any Western interference in the Syrian conflict.
The Americans reportedly have been pressing Idriss and his allies to go after the Islamists, particularly al-Nusra, the most disciplined and effective rebel group.
All-out war between the rival groups could be disastrous at this time since the wider rebel movement is facing growing pressure from regime forces in critical battles underway in Damascus and the northern city of Homs.
But, noted the U.S. global security consultancy Stratfor, "such a rift could greatly allay Western fears over arming the rebels.
"The U.S. Congress will be far more inclined to support the SMC if the council is actively fighting not only the Assad regime but also its one-time allies, the jihadists."
The National newspaper, published in the United Arab Emirates, reports that U.S. intelligence officers have been pressing "moderate Sunni leaders" to move against the front.
One rebel leader said that when he met U.S. agents in Jordan, where the Central Intelligence Agency reportedly funnels weapons to rebel forces and trains fighters, one even discussed drone strikes on al-Nusra camps to support attacks by SMC-affiliated groups.
"The U.S. intelligence officer said, 'We can train 30 of your fighters a month, and we want you to fight al-Nusra'," the rebel chieftain was quoted as saying.
"I'm not going to lie to you. We'd prefer you fight al-Nusra now, and then fight the Assad's army."
Stratfor noted that "Paris, London and Washington have been pushing the more moderate rebel groups, such as the SMC, to act against the jihadist groups.
"Though open strife between the rebel factions undermines the rebellion, the West is worried about the jihadist groups' growing strength and influence in Syria.
"The jihadists have assimilated many foreign fighters, including many from Europe, and there is considerable concern that the groups could seize chemical and other weapons, such as man-portable air defense systems, or send hardened militants back to the West."