Washington and Ankara are also at odds over the Syrian war, Turkey's plan to buy missile defenses from Russian Federation and the US conviction of a Turkish state bank executive on Iran sanctions-busting charges this year.
State Department spokeswoman Heather Nauert said Pompeo and Cavusoglu "had a constructive conversation" and "agreed to continue to try to resolve the issues between our two countries".
Pompeo met with Turkish Foreign Minister Mevlut Cavusoglu on the sidelines of a Southeast Asian security conference in Singapore.
Despite a sharp deterioration in relations over the detention of an American pastor, the United States and Turkey remain valued partners, U.S. Secretary of State Mike Pompeo said Saturday, in an apparent bid to ease tensions that have rocked ties between the North Atlantic Treaty Organisation allies.
The standoff appears to be one of the most serious crises between Turkey and the United States in modern history, along with the rows over the 1974 Turkish invasion of Cyprus and the 2003 US-led invasion of Iraq. He risks up to 35 years in prison if found guilty.
"Brunson needs to come home, as do all the Americans being held by the Turkish government", Pompeo told reporters before the meeting.
After reports that the U.S. would sanction individuals over the case, the lira hit a record low of 5.01 against the dollar, before rallying slightly to 4.9.
The Foreign Ministry said it would retaliate against what it called Washington's hostile action. Mr. Erdogan's government also alleges the Mr. Brunson has ties to a Turkish preacher now living in exile in the US whom Ankara accuses of fomenting the coup. They have been at odds over numerous matters, including military activity in northern Syria and Turkey's plans to purchase an advanced air defense system from Russian Federation.
Brunson, 50, is being tried on espionage and terror-related charges, which he and the USA government vehemently deny. On Tuesday, a court rejected his appeal to be released from house arrest during his trial on terrorism charges.
Last week, President Donald Trump and Vice President Mike Pence threatened to impose sanctions on Turkey if Brunson was not immediately released.
Mr Erdogan has previously tried to swap him for Fethullah Gulen, an Islamic cleric... He was also charged with supporting outlawed Kurdish PKK militants. Gulen denies the allegations.
Turkey has been trying to have Gulen extradited from the United States for two years.