Washington has imposed financial sanctions on two Turkish ministers and doubled steel and aluminum tariffs on Turkey as the Trump administration tries to secure the release of Brunson, who it says was framed.
A Turkish court also rejected the American pastor's appeal to be released from house arrest and have his travel ban lifted.
"We have put sanctions on several of their Cabinet members", Mnuchin said.
As a court in the western Turkish city of Izmir rejected a new appeal to free Brunson on Wednesday, spoksman Kalin said the U.S. needs to be respectful of the judicial procedures in Turkey. Turkey had previously said it would boycott USA electronic products.
"They are now holding our wonderful Christian pastor, who I must now ask to represent our country as a great patriot hostage".
Turkey's currency value has tanked since the United States sanctions were declared.
Berat Albayrak, who has faced criticism for failing to tackle the country's growing financial crisis, spoke to around 6,000 investors on a conference call to rebuff concerns that a funding squeeze on Turkey's banks and a damaging trade war with the United States would force him to seek a rescue bailout from the International Monetary Fund (IMF).
Among the big concerns is that Turkey's has amassed high levels of foreign debt to fuel growth in recent years.
The lira had already rebounded about 8 percent on Tuesday on news of a planned conference call on Thursday in which the finance minister will seek to reassure worldwide investors.
Albayrak rejected seeking a bailout from the International Monetary Fund, an option never seen as likely given that Erdogan often boasts of paying off Turkey's past International Monetary Fund debts in May 2013.More news: Bayer shares plunge after Monsanto cancer ruling
Despite the rise, Turkey's currency has lost nearly a third of its value against the dollar since January, pushing up the price of everyday items and raising fears that its weakness could infect other emerging market currencies.
French President Emmanuel Macron (L) meets Turkish President Tayyip Erdogan (R) in New York, US on September 19, 2017.
Trump has called Brunson's detention a "total disgrace".
Turkey retaliated with some 533 million dollar of tariffs on some USA imports - including cars, tobacco and alcoholic drinks - and said it would boycott U.S. electronic goods, singling out iPhones.
The lira's drop in value is certain to further fuel inflation, which is already near 16 per cent.
Some analysts said that USA tax cuts passed a year ago by Congress, which are boosting corporate earnings and economic growth, have reinforced the reasons for raising interest rates.
Michael Hewson, chief market analyst at CMC Markets UK, warned the Turkish tariff hikes risked provoking a new reaction from Trump and ultimately adding to downward pressure on the lira.
Qatar pledged $15 billion of investment in Turkey on Wednesday that a government source in Ankara said would be channeled into its banks and financial markets. Cavusoglu said ties with the bloc, long strained, were on a firmer basis and had started normalizing.
"If [the United States] have the iPhone, there's Samsung on the other side", he said, referring to U.S. giant Apple's iconic phone and the top South Korean brand. "This is very easy to resolve", the administration official said.