A prime task of President Donald Trump's top diplomat is to allay Turkish anger over US policy in Syria, a dispute which has ignited the biggest crisis in bilateral ties since the 2003 Iraq war.
But Turkish presidential sources said Erdogan made no secret of Turkey's concerns during the over three-hour meeting with the former chief of energy giant Exxon Mobil.
"In the talks, Turkey's priorities and expectations on all these topics were clearly conveyed to the US secretary of state," said a source, asking not to be named.
A State Department official said Tillerson and Erdogan had a "productive and open" conversation about a "mutually beneficial way forward" for US-Turkey ties.
Turkey's operation against the People's Protection Units (YPG) Kurdish militia in the Afrin region of Syria has added a potentially insurmountable new problem to an increasingly rough bilateral relationship.
Analysts said the level of tension was similar to 2003 when Turkey refused to let US troops operate from its territory for the Iraq war, or even the aftermath of Ankara's invasion of Cyprus in 1974.
Speaking in Beirut before his meeting with Erdogan, Tillerson denied Washington ever gave heavy weaponry to the YPG and thus could not gather up such arms, as desired by Ankara.
"We have never given heavy arms to the YPG so there is none to take back," Tillerson said.
Erdogan earlier this month accused Washington of sending in thousands of truckloads and planeloads of weapons to the YPG in Syria, asking why the US still had a presence there if the jihadists had been defeated.
He also warned that the US risked being dealt an "Ottoman slap" in Syria -- a backhand thwack which, according to legend, could kill an opponent at a stroke.
For Ankara, the YPG is a branch of the outlawed Kurdistan Workers' Party (PKK), which is blacklisted as a terror outfit by the US and the EU.
But for Washington, the YPG is an ally against IS jihadists and Turkey's operation is a distraction from efforts to ensure the extremists are permanently defeated.
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