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.
Turkish diplomats have told the USA government they are close to releasing Andrew Brunson, the imprisoned American pastor at the heart of the diplomatic row between the two countries, Middle East Eye can reveal.
"We have more that we are planning to do if they don't release him quickly", Mnuchin said during the meeting.
Brunson is under house arrest after spending almost two years in a Turkish prison under what the Trump administration has said are bogus allegations of aiding terrorists.
A Turkish court on Wednesday rejected a new appeal to free U.S. pastor Andrew Brunson, whose detention has sparked a major row between Turkey and the United States, local media reported.
Last week, Trump tweeted that his administration was doubling aluminum and steel tariffs for Turkey, and in response Ankara sharply hiked tariffs on some USA products.
Bilateral ties have spiraled into a full-blown crisis over the trial of Andrew Brunson, a Christian pastor, who is accused of having links with perpetrators of a failed military coup against Erdogan in 2016.
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In a conference call with thousands of investors and economists, Albayrak - who is President Tayyip Erdogan's son-in-law - said Turkey fully understood and recognized all its domestic challenges but was dealing with what he described as a market anomaly.
Turkey says the case is a matter for the courts.
Investors are concerned that Turkey has amassed high levels of foreign debt to fuel growth in recent years.
The dispute over his release has seen the two North Atlantic Treaty Organisation allies impose tariffs on one another's goods.
Mr Erdogan is angry that the United States has not taken more action against the Gulenist movement and what he said was a failure "to unequivocally condemn" the 2016 coup attempt.
Albayrak, a 40-year-old former company executive with a doctorate in finance, said Turkey would not hesitate to provide support to the banking sector.
Turkey's biggest mobile phone importing companies have cancelled their iPhone orders as Turkish citizens join a move to boycott USA products.
The currency gained some support from the announcement late yesterday of a Qatari pledge to invest $15 billion in Turkey.
The Turkish currency was trading at 5.85 at 1740 GMT, more than 1 percent stronger on the day.