US Secretary of State Mike Pompeo said Saturday he was "hopeful that in the coming days" Turkey would free Brunson.
Following the meeting, State Department spokeswoman Heather Nauert said that Pompeo "had a constructive conversation" with Cavusoglu and they " agreed to continue to try to resolve the issues between our two countries".
On Wednesday, Washington moved to block the assets of Turkish Justice Minister Abdulhamit Gul and Interior Minister Suleyman Soylu in the United States and prohibit American citizens from "engaging in transactions with them".
He gave an nearly two hour policy speech in Ankara Friday, without once referring directly to the tensions.
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The Turkish lira, which has been in decline against the dollar for most of 2018, fell to a record low of five to the dollar on Wednesday, after dropping 1.7 percent following the sanctions announcement.
The standoff appears to be one of the most serious fallouts between the two North Atlantic Treaty Organisation allies in modern history, along with the rows over the 1974 Turkish invasion of Cyprus and the 2003 US-led invasion of Iraq.
The Turkish lira on Wednesday slumped to record lows of 5.0 against the dollar as the United States hit Turkey's justice and interior ministers with sanctions over the case of an American pastor on trial for terror-related charges. He denies the charges but faces up to 35 years in jail. Pence, who has close ties to the evangelical Christian community, has been following proceedings since Brunson was arrested and pressing behind the scenes for action, aides said.
Mr. Brunson had denied all charges and his case has become a cause celebre for USA evangelicals and the Trump administration.
Pastor Brunson was arrested in Izmir, Turkey in October 2016, and was accused without any evidence of helping terrorist groups and of engaging in espionage.
President Recep Tayyip Erdoğan said Friday that he expects the roadmap in Syria's Manbij to continue with the USA, without being affected by the recent tensions.