The Obama administration quietly blessed the shipments to Libya of machine guns, automatic rifles, mortars and ammunition, but American officials later grew concerned as evidence grew that Qatar was giving the weapons to Islamic militants there.

American and Arab officials have expressed worry about something similar happening in Syria, where Islamists in the north have turned into the most capable section of the opposition, in part because of the weapons from Qatar. Saudi Arabia recently has tried to wrest control from Qatar and take a greater role in managing the weapons shipments to Syrian rebels, but officials and outside experts said the Qatari shipments continue. The greatest worry is over the shoulder-fired missiles — called man-portable air-defense systems — that Qatar has sent to Syria since the beginning of the year. Videos posted online show rebels in Syria with the weapons, including the Chinese FN-6 models provided by Qatar, and occasionally using them in battle.

The first videos surfaced in February and showed rebels wielding the Chinese missiles, which had not been seen in the conflict previously and were not known to be in Syrian government possession. .

Western officials and rebels alike say these missiles were provided by Qatar, which bought them from an unknown seller and brought them to Turkey. The shipment was at least the second antiaircraft transfer under the Qataris’ hand, they said. A previous shipment of Eastern bloc missiles had come from former Qaddafi stockpiles.

The shipments were small, the Western officials and rebels said, amounting to no more than a few dozen missiles. And rebels said the Chinese shipments have been plagued with technical problems, and sometimes fail to fire. The first FN-6s were seen in the custody of groups under the Free Syrian Army banner, suggesting that they were being distributed, at least initially, to fighters backed by the United States and not directly to extremists or groups with ties to Al Qaeda.

American and Arab officials said that Qatar’s strategy was a mixture of ideology — the ruling family’s belief in a prominent role for Islam in political life — and more hard-nosed calculations.

“They like to back winners,” one Middle Eastern official said.

In meetings with Mr. Obama, the leaders of Jordan and the United Arab Emirates have expressed a host of grievances about the Qatari shipments and have complained that Qatar is pursuing a reckless strategy.