Brunson, a 50-year-old evangelical pastor from Black Mountain, North Carolina, is being held on charges related to terrorism.
The currency plunged to a record low of 7.24 per dollar at the start of the week as a worsening of relations between Turkey and the United States added to losses driven by concerns over President Tayyip Erdogan's influence over monetary policy.
The currency has lost almost 40 percent against the dollar this year, driven by worries over Erdogan's growing influence on the economy and his repeated calls for lower interest rates despite high inflation.
President Donald Trump last week ramped up his attack on Turkey by doubling USA tariffs on Turkish aluminum and steel imports.
"We have more that we're planning to do if they don't release him quickly", Mnuchin told a cabinet meeting, without elaborating on what would be involved.
"The Brunson case is a rallying call for evangelicals within the Republican base because you have a Christian pastor being held in a Muslim-majority country", Amanda Sloat of Brookings Institution told the newspaper.
"They have not proven to be a good friend", Trump said of Turkey during the Cabinet meeting.
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Turkish President Recep Tayyip Erdogan has accused the U.S. of trying to "bring Turkey to its knees".
World stocks rose on Friday as news of plans for U.S.
Elmira Bayrasli, who teaches worldwide affairs with a focus on Turkey at Bard College, said that Erdogan is improving relations with Europe to increase confidence in the Turkish market.
Qatar has promised to make a Dollars 15 billion direct investment in Turkey, badly hit by a currency crisis amid a widening diplomatic standoff with the United States, officials said.
"We got somebody out for him", Trump said.
Dashing hopes for a quick solution to the dispute, a Turkish court on Friday rejected an appeal for the pastor's release from house detention. He is accused of having links to a group that Ankara says was behind the coup. Its leader lives in Pennsylvania and the USA has rejected calls from Turkey to have him extradited.
US President Donald Trump last Friday announced he would double tariffs on Turkish steel and aluminum, mocking the slide of the Turkish lira on Twitter. The lira's sharp decline has sparked fears in the global community of a Turkish economic crisis.
Erdogan thanked the emir and the people of Qatar for "standing by Turkey", the official Anadolu news agency reported.
This defiance seems to have consolidated Erdoğan's domestic power, but given the country's considerable economic reliance on Western banks and markets, the country is now more vulnerable than ever to a currency and debt crisis of its own making.