Brunson was moved to house arrest last week following almost two years in jail on terror-related charges but the change only increased tensions.
Also caught in the spat is a plan for the United States military to deliver new F-35 fighter jets to Turkey.
"The Turks were on notice that the clock had run and it was time for Pastor Brunson to be returned and I hope they'll see this for what it is, a demonstration that we're very serious", Pompeo said.
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. These are innocent people. "They agreed to continue to try to resolve the issues between our two countries", Nauert said.
Rep. Gabbard Attends Ceremony Honoring Return of Servicemen Remains
There are 9,569 United Nations Command personnel who remain unaccounted for from the Korean War, including 7,699 Americans. Numerous fallen service members died in North Korea and were buried by their comrades where they fell.
Cavusoglu, in comments to Turkish broadcasters, said he and Pompeo had also discussed potential joint steps the two countries could take regarding Syria's Idlib and Manbij.
Turkish President Recep Tayyip Erdogan says his country is facing an "economic war" after the lira hit a fresh record low against the dollar amid rising tensions with Washington.
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. The United States has also been seeking the release of three locally employed embassy staff detained in Turkey. His next hearing is set for October 12.
He is accused of carrying out activities on behalf of two groups viewed by Ankara as terrorists: the movement of the US-led Muslim preacher Fethullah Gulen blamed for the July 2016 coup attempt and the Kurdistan Workers' Party (PKK).
Erdogan saYS China, Mexico, Russia and India will be new markets for Turkey's exports.