It remains unclear how close the two sides remain, with some signs of movement but little indication of a shift in some of the key demands that have so far blocked any truce for more than a year.
Blinken's comments in the interview Saturday were not all that predictable when nearly 20 minutes of the 50-minute conversation turned toward Israel, Hamas, and Gaza.
An additional impediment to a deal was Hamas's wish to see the conflict expanding to additional fronts, Blinken noted.
If Hamas truly cares about the Palestinian people, they will "they will say yes and do it now," Blinken said.
Blinken said at a news conference that the rebels had agreed on a joint communique that also calls for an inclusive and representative government.
A US official told Reuters that Washington sees this moment as an opportunity to further push back Iran's influence in the region.
Blinken's conversation with the Turkish diplomat comes in the midst of his Middle East tour to bolster a united front with Arab and Turkish allies in the region.
He added that possible Turkish and US roles for Syria's future were among the topics they discussed.
Blinken calls Assad’s fall a turning point, urging rights protection and a Syrian-led transition.
Blinken said he would not speculate about "what's going to happen in the future" and that he can only talk about the clear interests of the parties.
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