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Asked about the German report, one senior U.S. intelligence official told The Times that “it would be inaccurate to suggest the government of Turkey is helping foreign jihadists enter Syria.”

“Preventing extremists from crossing the roughly 500-mile Turkish border to join the Syrian opposition is among the many security challenges Turkey faces,” said the official, who spoke on the condition of not being named in this article.

A Turkish government source, who also asked not to be named, said the U.S. comment was unsurprising “because American authorities are very well aware of our operations against al Qaeda.”

“To claim that Turkey is supporting terrorism is an insult to Turkey,” said the Turkish source. “We’re trying to control our borders as much as we can.”

Others say the situation is complex and nuanced by Turkey’s domestic and regional political strategies.

Hugh Pope, an Istanbul-based program director for the International Crisis Group, said the Erdogan government may have deliberately blurred its border with Syria in order to provide aid for hundreds of thousands of refugees crossing into Turkey from Syria, but dismissed the notion that the Erdogan government supports al Qaeda-linked fighters crossing the other way.

“This is certainly not Turkey embracing al Qaeda,” Mr. Pope said in an interview with The Times. “Al Qaeda has blown plenty of stuff up in Turkey. Turkey is an enemy to al Qaeda and has been a partner to the United States in aiding the fight against al Qaeda in all kinds of ways.”

Paul Pillar, a former CIA officer who now teaches at Georgetown University, said that “if the Erdogan government is relaxed about aid getting into the hands of Nusra, it’s because they see Nusra as one of the more effective fighters against Assad and not because this is some kind of backdoor way of aiding whatever particular ideology Nusra represents.”