Brunson, an evangelical Christian pastor originally from North Carolina but a longtime resident of Turkey, was accused by the Turkish government of terrorist involvement and playing a role in the Turkish military's 2016 attempted coup to overthrow the government.
"We have more that we are planning to do if they don't release him quickly", Mnuchin said during a meeting of President Donald Trump's Cabinet.
But according to Middle East Eye, Turkish diplomats said they could release Brunson anyway, subject to further concessions from the United States which could soothe diplomatic tensions and helping alleviate the diplomatic and economic crisis.
Trump prefaced Mnuchin's remarks by saying that Turkey had not been a very good friend.
"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.
US stocks suffered their worst day in seven weeks after the move, and the lira gained 7 percent against the dollar. "Our relations with Turkey are not good at this time!"More news: Chad Kelly’s Debut Creates QB Conundrum In Denver
"It nearly seems as if Trump's words were an invitation for there to be even more of an entrenchment of this dispute", said Al Jazeera's Sonia Gallego, reporting from Turkey's largest city of Istanbul.
Albayrak, Erdogan's son-in-law, told investors on Thursday that Turkey would emerge stronger from the currency crisis, insisting its banks were healthy and signaling it could ride out the dispute with Washington.
But while relations with the European Union are improving, there are no signs so far that Turkey or the US are willing to back down.
Mr Trump wrote in a tweet late on Thursday: "We will pay nothing for the release of an innocent man, but we are cutting back on Turkey!"
On Monday, the central bank said it would provide all the liquidity Turkish banks needed, as it seeks to keep money flowing in the financial system.
The currency crisis has deepened concerns about the broader economy - particularly Turkey's dependence on energy imports and whether foreign-currency debt levels pose a risk to the banking sector.
Kristina Arriaga, vice chair of the US Commission on International Religious Freedom, recently returned from Turkey.
On Friday, Turkish Trade Minister Ruhsar Pekcan said her government would respond to any new trade duties imposed on the country, whose currency has plunged in recent weeks. Inflation is spiralling (currently more than 15%), Turkish companies are saddled with foreign debt and the country has one of the world's largest current account deficits in proportion to its economic output, heightening fears of a debt crisis.