Story highlights U.S. Central Command says it's not aware of ISIS conducting any flight operations

ISIS has three warplanes in Syria, a London-based Syrian opposition group claims

Former Iraqi military officers are training members of ISIS to fly the planes, it reports

ISIS has captured large swaths of northern and eastern Syria, as well as parts of Iraq

Could ISIS fighters be training as warplane pilots capable of turning their weapons on coalition aircraft?

That's the worrying prospect suggested by an unverified report from the London-based Syrian Observatory for Human Rights that ISIS has three warplanes in its possession in northern Syria.

The Syrian opposition group also claims, citing reliable sources in Syria, that former Iraqi military officers are training members of the Sunni extremist militant group to fly them.

The planes are at the Al-Jarrah military airport east of the contested city of Aleppo, the Syrian Observatory for Human Rights said.

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The former Iraqi military officers are among those who have joined ISIS, and people living near the airport have said that they recently saw an airplane flying at low altitude after it took off, according to the Observatory.

Information on how the Islamic State in Iraq and Syria, which calls itself the Islamic State and is also known as ISIL, obtained the planes, or the trainees' previous level of familiarity with flight, wasn't immediately available.

CNN has not independently verified the report.

However, the U.S. Central Command sought to pour cold water on the report.

"We're not aware of ISIL conducting any flight operations in Syria or elsewhere," said spokesman Col. Patrick Ryder.

"We continue to keep a close eye on ISIL activity in Syria and Iraq and will continue to conduct strikes against their equipment, facilities, fighters and centers of gravity, wherever they may be."

ISIS has captured large swaths of northern and eastern Syria, as well as parts of western and northern Iraq, for what it says is its new Islamic caliphate.

Kobani airstrikes target ISIS

This week has seen an increased number of U.S. airstrikes against ISIS in the northern Syrian city of Kobani, about 90 miles (145 kilometers) northwest of Aleppo.

The strikes are helping Kurdish fighters on the ground push back ISIS militants in the Syrian town, according to sources there.

The Pentagon said its increased focus on the area reflected greater opportunity rather than a strategic shift with regards to the importance of helping the Kurds hold Kobani.

"What makes Kobani matter for us from an airstrike perspective is that (ISIS is) there, and that they want it," said Rear Adm. John Kirby, a Pentagon spokesman.

Retired Brig. Gen. Mark Kimmitt told CNN Friday that Kobani is not tactically or militarily very important -- but has taken on a certain significance.

"It's a small little town along the border with Turkey, he said. "However, it has attracted the attention of the world.

"The world is watching the coalition fight ISIS and, because of that, a victory by either side in Kobani will have a psychological and strategic communication victory that far outweighs the tactical military victory to whichever side wins."

In recent weeks, administration and U.S. defense officials have said it wasn't essential to keep the city, on the border with Turkey, from becoming the latest in Syria to fall into ISIS hands.

Anbar advances

Meanwhile, in Iraq's western Anbar province, ISIS militants have tightened their grip over an area about 15 miles south of Falluja, a provincial official said Thursday.

ISIS has brought many of its fighters, armed with heavy weaponry and machinery, in order to attack and control the town of Amiryat al-Falluja, the head of the town's council, Shaker Mahmoud, said Thursday.

Mahmoud urged the Iraqi government to send military reinforcements to support the security forces inside the town.

The deputy head of the Anbar provincial council, Falleh al-Issawi, told CNN by phone Friday that a curfew has been imposed on the city of Ramadi until further notice.

This was prompted by fear of potential suicide attacks by ISIS fighters, particularly during Friday prayers, he said. The curfew was imposed Thursday night.

Al-Issawi told CNN the situation in Ramadi was critical, with "heavy clashes between Iraqi security forces and their supporters from one side and the ISIS militants from another."

Car bomb attacks claimed by ISIS targeted Shia areas of Baghdad Tuesday, killing at least seven people and injuring more.