Global stocks also suffered this week as the sell-off spilled into developed markets, adding to the angst over U.S.
Brunson, an evangelical preacher from North Carolina, is now under house arrest in Izmir.
The Middle East Eye reported that Turkish diplomats are ready to comply - but want something in return.
In recent weeks, the U.S. has sanctioned a pair of Turkish cabinet members, and dramatically increased tariffs against the nation, fueling anger among Turkish leaders and creating economic uncertainty across the region. He has been held in Turkey for two years and accused by local prosecutors of "terrorist" ties to the Islamist Gulen movement, which has been blamed for a failed coup in July 2016.
Erdogan previously suggested a swap in which Brunson is released in exchange for Gulen, though USA officials have said Turkey must present convincing evidence for any extradition proceeding to be considered.
The court ruling comes amid a diplomatic spat between Turkey and the U.S. that has seen tariffs from both sides pile up and an exchange of threats of further sanctions.
The Eye's report is based on comments from an anonymous diplomatic source.
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Inflation in the state is at a 14-year high in Turkey, at close to 16%, but the country's central bank - which is under the control of Erdogan - refuses to raise interest rates, a simple mechanism for bringing down rampant inflation.
"Turkey will emerge stronger from these (currency) fluctuations", Albayrak was quoted by state-run TRT television as saying during the conference call.
The bail-out followed talks in Ankara between the emir of Qatar, Sheikh Tamim bin Hamad Al-Thani, and Turkey's President Recep Tayyip Erdogan.
Brunson, from North Carolina, is under house arrest as his trial on terrorism charges continues.
In a post on his Twitter page, Gargash referred to a recent visit to Ankara by the Qatari emir and his pledge to help Turkey overcome an ongoing currency crisis and described it as an attempt to "buy sovereignty".
Turkey has sought to persuade the United States to spare Turkey's state-owned Halkbank from a threatened fine for allegedly helping Iran evade USA sanctions.
"I have great faith in the American people and I'm hoping that at a minimum, anyone who has a Nutella sandwich, anyone who has towels that come from Turkey that unwillingly they're participating in lifting the economy of Turkey, which is something that we should not be doing when they're holding Americans like Pastor Brunson", Arriaga said.
The White House has said there could be more economic measures against Turkey if Brunson is not released.
The worsening political relationship between Turkey and the USA does not help.