"Naturally, we welcomed these statements", said Cavusoglu, adding: "Of course we want to see this put into practice".
Cavusoglu again repeated that Turkey saw the YPG as a "threat" trying to divide Syria.
Trump and Erdogan spoke on the phone earlier on Friday and discussed recent developments in Syria, bilateral ties and a summit in Russia's Sochi, sources in Erdogan's office said.
After Assad's father, Hafez al-Assad, expelled Syrian Kurdish nationalist leader Abdullah Ocalan in 1998, Syria and Turkey maintained relatively warm relations until 2011 when protests against the younger Assad transformed into armed rebellion against the state.
USA -backed Kurdish and Arab militias seized the city in October.
The PKK - blacklisted as a "terror" group by Turkey and its Western allies including the U.S. - has waged an insurgency inside Turkey since 1984.
U.S. President Donald Trump on Friday held a phone conversation with his Turkish counterpart Recep Tayyip Erdogan, pledging that Washington will not provide weapons to the People's Protection Units (YPG), Turkish Foreign Minister Mevlut Cavusoglu said.
Joe Robinson was arrested while on holiday in Turkey, accused of fighting alongside Syrian Kurdish militia against the Islamic State group in Syria.
Erdogan was speaking on board his plane on his return from a trilateral meeting with Russian Federation and Iran to promote a peaceful settlement in Syria that took place in the Black Sea resort of Sochi on Wednesday. But instead, the U.S.is not only sticking with the Kurdish YPG in Syria; they're going to directly arm them - small arms, reportedly, but definitely not what Turkey was hoping to hear. The Syrian military and Syrian Kurdish forces shared control of the northern Syrian town of Manbij in March to block an oncoming advance of Turkish-backed Syrian Arab rebels. The YPG was a key US ally against the Islamic State group in Syria.