The rebels' claim has not been independently verified and Syria's state news agency attributed the crash to a technical error, The New York Times reported.
A video of the incident uploaded to YouTube shows a jet rushing across the screen before bursting into flame. It is described as "the downing of a MIG-23 by the Grandsons of Mohammed Brigade."
Rebels and activists shared the video widely online and have called it a great achievement for the opposition, which has been outgunned by President Bashar Assad's army throughout most of the country's now 17 months of fighting.
"It's a big operation, a big blow to the regime," said Abu Rawan, a commander with the Free Syrian Army in Homs province. "It means the [Free Syrian Army] is strong and because we can down these kinds of planes, we can institute a no fly-zone without international support."
Meanwhile, the army's bombardment of Aleppo, Syria's largest city and commercial center, focused on the opposition Salahuddin neighborhood, part of an arc of rebel-controlled territory stretching around the Chicago-size city from the northeast to the southwest.
Air attacks Sunday also included barrages of several other rebel-held neighborhoods, opposition activists said. The barrages included so-called air bursts, when some of the shells explode in the air above the target instead of upon impact.
Assad's regime, which calls the rebel fighters terrorists, also cut off landline phone service, Internet and electricity in many rebel-held neighborhoods, rebels said.
Rebels and regime forces also clashed in street battles, especially in Salahuddin, which rebel commanders said could prove decisive in the struggle for control of the city.
"If Salahuddin falls, it will give us control over the Damascus-Aleppo highway, and it will cut off the regime's logistics pipeline into the city," rebel field commander Abdel Qader Saleh, 32, told The Wall Street Journal.
He said rebel defensive gains and their ability to hold territory were transforming their guerrilla campaign into a more traditional conflict, with a shifting but defined front line.
Saleh and other rebels fighting in Aleppo told the Journal they were angry the West was unwilling to offer substantive support.
He ridiculed a Friday pledge by British Foreign Secretary William Hague to provide $7.8 million of communications equipment, body armor and medical supplies to rebel forces.
"My Nokia works just fine," Saleh said. "I don't need walkie-talkies. That's like giving a dying man sunglasses."
He said his 8,000 fighters had received just two shipments of weapons from Turkey -- one of 300 rifles and rocket-propelled grenades, and a second with about 3,000 hand grenades.
A Turkish official said Sunday Turkey discussed the possibility of setting up a no-fly zone over Syria with U.S. Secretary of State Hillary Clinton, who met with her Turkish counterpart in Ankara Saturday. But no decision was reached, he said.
The imposition of a no-fly zone by foreign powers is widely believed to have been crucial in helping Libyan rebels overthrow dictator Moammar Gadhafi last year.
White House press secretary Jay Carney said the Obama administration "have ruled out no options as we try to bring about, with all of our partners, and with the Syrian people, the political transition that is so desperately needed in Syria."
The course the United States and its partners still believe the course they are taking "in collectively putting pressure on the Assad regime, isolating the Assad regime and depriving it of resources to continue its attacks against the Syrian people, is the right course," Carney said.
"But we review all options," he said, "and will continue to do so."
Meanwhile, the United Nations said the violence, the use of heavy weapons by the Syrian government and targeted attacks by the opposition are taking a heavy toll on innocent civilians
"It is clear that violence is increasing in many parts of Syria," U.N. Military Adviser Lt. Gen. Babacar Gaye, head of the U.N. Supervision Mission in Syria, said Monday in a release. "Our patrols are monitoring the impact of this violence, visiting internally displaced people and hospitals."
The United Nations said an estimated 17,000 people, mainly civilians, have been since the uprising against Assad began in March 2011.