"We have more that we are planning to do if they don't release him quickly", Mr Mnuchin said during the meeting.
The US present had harsh words for Turkish officials during the cabinet session, saying, "They have not proven to be a good friend".
The White House stance appeared to give Turkish authorities little incentive to work for the release of Andrew Brunson, a pastor on trial in Turkey on terrorism charges and whose case Turkish officials have said was a matter for the courts.
The threat of sanctions comes amid Trump's concerns over Brunson, who is accused of being involved with an attempted coup that led to tens of thousands of people being arrested in 2016.
In the currency market, the lira bounced back to 5.815 per dollar, up nearly 25 percent from its record low of 7.2400 hit early on Monday, despite threats of more sanctions from Washington.
Investors said Albayrak's conference call would test whether Turkey can persuade markets that its monetary policy is not hostage to political influence.
Before he spoke, the lira strengthened more than 3 per cent, despite signs that the rift with the United States is as wide as ever. "Not fair. Not right".
Virat Kohli loses his number 1 Test ranking to Steve Smith
Woakes (2/24) continued his excellent performance trapping Hardik Pandya (26) lbw before Anderson came to clean up the tail. After being bowled out for 107 runs in the first innings, they were folded to 130 in the second innings.
Global stocks also suffered this week as the sell-off spilt into developed markets, adding to the angst over US-China trade relations and the fiscal prudence of Italy's anti-establishment government.
Turkey shot back on Friday, threatening to respond if the United States levied further sanctions. Referring to imprisoned pastor Andrew Brunson, Trump said "they have a great Christian pastor there, he's a very innocent man".
Ankara stood by Doha after Saudi Arabia and other Arab states severed diplomatic, trade and travel ties with Qatar previous year, accusing it of financing terrorism, a charge Doha denies.
The two North Atlantic Treaty Organisation allies have traded tit-for-tat sanctions over the pastor's house arrest.
The finance minister will seek to reassure worldwide investors on Thursday in a conference call for which a ministry official said some far 3,000 people had signed up.
Turkey's biggest mobile phone importing companies have cancelled their iPhone orders as Turkish citizens join a move to boycott U.S. products.
The lira gained some support from the announcement late on Wednesday of a Qatari pledge to invest US$15 billion in Turkey.
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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.