But Ankara regards the YPG as a terror group and the Syrian branch of the Kurdistan Workers' Party (PKK), which since 1984 has waged an insurgency inside Turkey leaving tens of thousands dead.
However, the Syrian Kurdish group labelled a terror organisation by Turkey hailed the U.S. decision to arm it with heavy weapons to fight IS.
Turkish media are quoting Deputy Prime Minister Nurettin Canikli as saying the USA decision to arm Syrian Kurdish fighters viewed as terrorists by Turkey is "unacceptable".
Turkish President Recep Tayyip Erdogan on Wednesday called on the country's allies to "side with Ankara and not with terrorist organisations".
The recent USA decision to supply weapons to "PYD and YPG elements" - considered by Ankara as terrorist group PKK's Syrian offshoot, and its armed wing - will neither benefit the US nor the region, Turkish defense minister said Wednesday.
The YPG forms the backbone of the Syrian Democratic Forces, which has driven the IS group from much of northern Syria with the help of US air support.
The US insists it is not arming elements hostile to its North Atlantic Treaty Organisation partner. "I hope that they will turn away from this wrong", he said.
Mr Yildirim said he could not imagine the U.S. having to choose between Turkey's strategic partnership and a terrorist organisation.
De Mistura spoke ahead of a planned resumption of United Nations -mediated talks between the Syrian government and opposition next week in Geneva. Its leaders have railed against any strategy that, in their view, would involve the US using one terrorist group to fight another.
President Erdogan is scheduled to meet with President Trump next week in Washington and the decision to arm the YPG adds to the tensions. Talal Sillo, a commander with SDF, said no weapons have been delivered yet but noted the US -led coalition said a delivery was expected from weapons already in Syria with the global coalition.
Mr Erdogan has not yet responded to Mr Trump's decision, but has repeatedly castigated Washington for its support of the YPG.
"This is yet another victory by the SAC and the SDF, our most committed and capable ground force partners in the fight against ISIS who remain hard at work erasing ISIS from the battlefield, liberating their own people and lands", said the Coalition Spokesman Col. John Dorrian.
Their half-hour conversation took place before an global conference on Somalia, and Mattis said he had no doubt the US and Turkey would "work this out with due consideration and significant attention paid to Turkey's security, to NATO's security and the continuing campaign against ISIS".
Information for this article was contributed by Sarah El Deeb, Lolita C. Baldor and Bob Burns of The Associated Press.