“They gave us roses at the airport, brought our bags, everyone wanted to take selfies with us,” Mr. Snyder said. “This was the best tournament I have ever participated in, even better than the Olympics in Rio.”

Later on Saturday, during a sightseeing trip, the wrestling team took to the landmark Milad Tower in Tehran, and excited Iranians took pictures of the team.

Jordan Burroughs, an Olympic champion, said he has more fans in Iran’s capital “than from any other city.”

For a time, it had seemed the American wrestlers would be forced to stay home. After President Trump announced that Iran would be included in a travel ban, the country decided it would reciprocate by barring the team. But the Iranian authorities changed their minds after seeing numerous protests in the United States against the ban.

Mr. Burroughs said that he was disappointed when he heard of the issues with the travel ban. “I don’t necessarily agree with all decisions by my government, and clearly not with this one,” he said of the now-halted order barring travel by Iranians and citizens of six other predominantly Muslim countries to the United States. “I have never been affected by terror, but here in Iran, I have never felt any ill will toward me — the opposite actually.”