The Syrian Observatory for Human Rights said the air strikes targeted Islamic State fighters near the town of Tel Tamr, where the militants, also known as ISIS, had captured 10 Assyrian villages.
A prominent Syrian Christian, Bassam Ishak, told Reuters: "Some people have tried to call them by cellphone, the relatives that have been abducted, and they get an answer from a member of ISIS who tells that they will send the head of their relative.
"They are trying to terrorize the parents, the relatives in the Christian Assyrian community," said Ishak, who is president of the Syriac National Council of Syria.
Islamic State has staged mass killings of religious minorities, as well as fellow Sunni Muslims who refuse to swear allegiance to the 'caliphate' it has declared in parts of Syria, Iraq and other areas of the Arab world.
Its fighters were shown beheading 21 Egyptian Coptic Christians in Libya last week. Last August, it killed or enslaved hundreds of Iraq's Yazidis, whom it considers devil worshippers, just over the nearby frontier in Iraq.
The militants have previously used kidnappings to trade captives for their own captured fighters. It was not clear if they planned to do the same this time.
"What is happening is a threat to our existence," said Ablahd Kourieh, an Assyrian Christian who is deputy head of a Kurdish-led defense council in northeastern Syria, speaking to Reuters from the region via Skype. He estimated the number of abducted Assyrians was even higher, at between 350 and 400.
He called on the U.S.-led alliance to mount air strikes and to arm Kurdish-led forces which are battling Islamic State in the region. He estimated that 3,000 Assyrians had fled from the villages for the main cities of Qamishli and Hasaka.
"We call for bombardment of the terrorists' positions there, and the provision of quality weapons," said Kourieh. He said there had been no contact with Islamic State, which has yet to claim the abductions.
The United States and its allies have carried out hundreds of air strikes in both Iraq and Syria since launching a campaign to "degrade and destroy" Islamic State last year. Washington on Wednesday condemned the attacks against Christians, which it said included the burning of homes and churches and abduction of women, children and the elderly.
ISLAMIC STATE "NEEDS MONEY"
The attack on the Assyrians coincided with a big offensive by Kurdish YPG fighters against Islamic State in northeast Syria, a strategically vital region because it borders territory the militants control in Iraq.
The Kurds say they have cut a main Islamic State supply line from Iraq. The Observatory says 132 Islamic State fighters have been killed in that battle, the latest of a recent string of setbacks for the group in Syria.
Rami Abdulrahman, who runs the Observatory, reported new signs of financial strain for Islamic State: it had started salvaging and selling scrap metal from bombed factories and railway lines in the eastern province of Deir al-Zor.
"They need money. Ever since the air strikes hit their oil facilities and the Turkish border has been harder to cross, they have increased taxes and looked for ways to make money," he said.
Abdulrahman said 35 Islamic State fighters had been killed in battles with the Kurdish YPG near Tel Tamr. The death toll in the YPG and allied Christian militia was 25, he said.
The abducted Assyrians are believed to have been taken into Islamic State-controlled territory in Hasaka.
IS ADVANCES NEAR DAMASCUS
While Islamic State has come under pressure in its strongholds in northeast Syria, its fighters have opened a new front hundreds of miles to the southwest near Damascus, the seat of President Bashar al-Assad's government.
The group said on social media its fighters had attacked rival insurgents east of Damascus in the Ghouta region, saying it had ambushed members of the Free Syrian Army, a name adopted by an array of mainstream rebel groups.
It said the FSA suffered casualties, and Islamic State fighters seized tanks and ammunition. The Observatory confirmed the attack but said it had targeted rival jihadist brigades.
Mainstream rebels fighting Syrian government forces and allied militia in Syria's civil war have mostly been eclipsed by jihadists, complicating a U.S. plan to train and equip Syrian opposition forces to fight Islamic State.
Units from Islamic State are staging increasing attacks in the south, mostly in the Qalamoun Mountain range that runs north-south to Damascus and also borders Lebanon.
The Lebanese army deployed along that border on Thursday and fired artillery at jihadists who move between Syria and Lebanon, a Lebanese security source said.
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By Oliver Holmesand Tom PerryAdditional reporting by Dasha Afanasieva in Istanbul; Editing by Mark Trevelyan. http://www.reuters.com/article/2015/02/26/us-mideast-crisis-syria-christians-idUSKBN0LU0KX20150226?