In testimony before Congress this week, the F.B.I. director, James B. Comey, said the challenge for F.B.I. agents was determining when someone has crossed the line from speech to criminal activity. “It’s even protected speech to say I’m a fan of the Islamic State so-called,” Mr. Comey said.

When the suspect is a cleric, like Imam Bengharsa, the matter is especially delicate.

“It’s very possible that he’s never crossed the legal threshold,” said Seamus Hughes, deputy director of the Program on Extremism at George Washington University, who has closely followed the imam’s story. But Mr. Hughes called the situation “perplexing and concerning.” The imam “can take his supporters right up to the line. It’s like a making a cake and not putting in the final ingredient. It’s winks and nods all the way.”

Imam Bengharsa appears to have plenty of money. Court records say he received $902,710 in wire transfers in 2014 and 2015, possibly an inheritance. He told The Detroit News that he often helped needy people like Mr. Gregerson. “If that individual turns around and wants to use that money for something else that’s illegal, the person who gave the money cannot be held responsible,” Imam Bengharsa said. “It’s pathetic if they are making those connections. If that’s what this country has become, I’d rather be in jail.”

The documents say he transferred money three times to an unnamed person in Yemen.

Investigators are also exploring contacts between Imam Bengharsa and other people suspected of extremism or terrorism. One is Yusuf Wehelie, 25, a Virginia man arrested in July and charged with weapons possession, which would be illegal because he has a previous felony conviction for burglary.

Mr. Wehelie first came to public attention in 2010, when he and his brother, Yahya Wehelie, both American citizens, were temporarily detained in Cairo and prevented by the F.B.I. from flying home. American officials said such delays were sometimes necessary to assess whether a person posed a security threat. The American Civil Liberties Union and the Council on American-Islamic Relations protested that the rights of such travelers were being violated.

Image Yusuf Wehelie Credit... Jacquelyn Martin/Associated Press

At Yusuf Wehelie’s detention hearing in July, the authorities said he had told undercover agents that he supported the Islamic State and that if he couldn’t join it overseas, he would attack a military recruiting center, possibly using explosives. (Mr. Wehelie’s lawyer, Nina Ginsberg, said that in later recorded conversations, he disavowed those statements and later stopped replying to the undercover agents.)