Presidential spokesman Ibrahim Kalin said Trump had heard Turkey's demands on establishing a safe zone in northern Syria plus the extradition of U.S.-based Fetullah Gulen.
Gulen is wanted by Turkey for his role in the Fetullah Terrorist Organization (FETO), blamed for the July 15, 2016 coup attempt.
In an interview with Turkish private broadcaster NTV, Kalin also said Syria, the PKK and PYD were discussed by the two leaders.
Kalin said establishing a safe zone in northern Syria was a reasonable solution to prevent a new wave of refugees. He said the Trump administration was also positive on the safe-zone issue.
"What we have foreseen is that the Azez-Jarabulus-Al Bab line is extended to an area of 5,000 square kilometers and declared a safe zone free from terrorism. They [U.S.] expressed positive views on that," Kalin said.
Kalin also said protecting Syria's territorial integrity was Turkey's basic principle for solving the issue.
The presidential spokesman said Turkey would continue to discuss these issues with the new administration in Washington and other countries in the region.
Though Erdogan has long favored a safe zone, the idea has made little headway with the powers engaged in Syria. However, last month Trump told U.S. broadcaster ABC news he "will absolutely do safe zones in Syria".
Kalin also criticized the administration of former U.S. leader Barack Obama for not taking steps on the extradition of Gulen.
He added the Trump administration was more positive on this issue.
"Mr. Trump also stated that they took the issue seriously and they would take necessary steps immediately; of course we [Turkey] will follow that closely," Kalin added.
Kalin also said Erdogan and Trump agreed to meet face to face soon.