In a swipe at Barack Obama during his final days in office, President Erdogan said he hoped a “consensus” would be reached with the President-elect that would allow both sides to solve “regional issues”.
While not elaborating on what they may be, it is thought he could be referring to the war in Syria and US support for a Kurdish militia group fighting Isis.
Diplomacy between the US and Turkey has suffered as a result of the six-year conflict and America’s support for the Kurdish People's Protection Units – or YPG – which Turkey sees as an extension of the militant group PKK.
"I believe we will accelerate dialogue when Mr Trump takes office," the Turkish leader told a conference of Turkish ambassadors.
"I believe we will reach a consensus with Mr Trump, particularly on regional issues."
Shifting alliances in the Trump era could have wide-ranging implications for US and Russian foreign policy in the Middle East.
Mr Erdogan’s comments come as a senior member of his government, Foreign Minister Mevlut Cavusoglu, also welcomed the incoming US president.
Mr Cavusoglu said the property tycoon was likely to extradite Fethullah Gulen, an Islamic cleric accused of organising a failed coup in Turkey last summer.
Mr Gulen has lived in self-imposed exile in Pennsylvania since 1999 and denies any involvement in the plot to oust Mr Erdogan in July.
Elaborating on Mr Trump, the Turkish foreign minister said he believed the President-elect would not make the same mistakes as the outgoing US administration in relations with Turkey.
Turkey expects Washington to extradite Gulen and to end its cooperation with the YPG, he said.
However, US officials say Gulen's extradition is a matter for the courts and that that process could take years.
Mr Trump's policy on the Kurdish militia remains unclear.
The West would lose a major ally in the war against Isis if Mr Trump were to cut support for the YPG, who have been instrumental in defeating the jihadi group in Syria.
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