Asked if the issue threatened Turkey's membership of the North Atlantic Treaty Organization (NATO), Pompeo said: "Turkey is a NATO partner whom the United States has every intention of continuing to work cooperatively with".
Speaking to Turkish journalists after the meeting, Cavusoglu also described the discussion with Pompeo as "constructive", but added: "Nothing can be achieved through threatening language and sanctions".
"I am hopeful that in the coming days we will see that occur", Pompeo said.
The NATO allies are also at odds over the Syrian war, Turkey's plan to buy missile defences from Russian Federation and the US conviction of a Turkish state bank executive on Iran sanctions-busting charges this year.
On the sanctions, he said: "I hope they'll see this for what it is, a demonstration that we're very serious".
Brunson, 50, was arrested in 2016 on charges of "committing crimes on behalf of terror groups without being a member" and espionage.
Relations between the United States and Turkey have plummeted over Brunson, who was in custody for 21 months in a Turkish prison until he was transferred to house arrest last week.
"We call on the USA administration to walk back from this wrong decision", it said in a statement on Wednesday.
One is led by US-based Turkish preacher Fethullah Gulen, whom Ankara blames for masterminding the failed 2016 coup against the Turkish government.
"Brunson needs to come home, as do all the Americans being held by the Turkish government". "President Trump has made it abundantly clear that the United States expects Turkey to release him immediately".
He was accused of helping supporters of Fethullah Gulen, a us -based cleric who Turkish authorities say masterminded the coup attempt in which 250 people were killed.
It added that the US' "aggressive attitude serves no objective", and vowed that it would respond in equal measure. Gulen denies the allegations.
ESPN's Paul Finebaum on what's next for Urban Meyer
The Ohio-based restaurant chain said in a statement it would be suspending its "head breakfast coach" partnership deal with Meyer. The extension also increased Meyer's salary to $7.6m for 2018, an $800,000 raise over what he was scheduled to earn.