"I think now we're as close as we've ever been", he said, as their hour-long meeting at the Lotte New York Palace Hotel began at 19:15 GMT.
This confrontation with American protesters - it should be noted - took place on the same day that Erdogan met with Trump on the sidelines of the UN Assembly.
"Washington should not harbor terrorists and they should be extradited to Turkey", Erdogan said.
"It is a great honor for me to introduce you President Erdoğan of Turkey, he has become a friend of mine", he said. "All of Iraq's neighbors, and virtually the entire worldwide community, also oppose this referendum".
Turkey's President Erdogan met with his American counterpart in a closed-door meeting on the sidelines of the UN General Assembly amid differences on United States policy on Syria, as well as a number of other issues.
Turkey has brought forward a cabinet meeting and national security council session to Friday over the referendum, Erdogan said. Erdogan also has detained or fired thousands of people under suspicion they were part of the coup attempt.
But the United States had already quietly put the sale on hold after the violence, and the Trump administration had informed the Turkish government that the sale wouldn't be allowed to take place.
At a critical time for Turkish-Germany relations, Erdogan said in the interview he hoped relations with Germany could be improved, praising German Chancellor Angela Merkel for refraining from criticizing Turkey and its policies.
The EU and human rights organisations have expressed concern over Turkey's security crackdown since the failed coup.
Turkish-American relations have also been strained over United States policy on Syria.
Erdogan also said the United States should extradite US -based cleric Fethullah Gulen, whom Erdogan accuses of orchestrating a failed coup in Turkey.
Turkey has requested Gulen's extradition from the USA, but the U.S. has so far failed to comply.