Trump said Brunson "is suffering greatly".
"If Turkey does not take immediate action to free innocent man of faith and send him home to America, then the United States will impose significant sanctions until Pastor Brunson is free", said Pence during his address.
The Secretary of State had been involved in delicate negotiations in an attempt to free the 50-year-old pastor, who had been jailed for a year and a half on what the US maintains are false charges connected to a failed coup attempt against Turkey's President Recep Tayyip Erdogan.
Andrew Brunson, an Evangelical pastor from North Carolina who has done missionary work in Turkey for 23 years, has been the subject of calls from President Donald Trump and other US government officials for his release.
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On Wednesday, a Turkish court ordered that Brunson be freed from prison and placed under house arrest while his trial continues, according to Brunson's lawyer.
Turkey and the United States engage in a war of words over Ankara's detention of an American pastor on terrorism charges.
If also convicted of espionage, Mr Brunson - who denies all charges - could receive another 20 years in jail.
Trump and Pence have had multiple conversations with officials in the Turkish government regarding the detained American pastor in recent weeks, an administration official tells CNN.
The threat marked a significant escalation in already tense relations between the two North Atlantic Treaty Organisation allies over a range of issues - from Syria to Turkey's purchasing of advanced Russian air defence systems - that have seen the governments grow further apart.
Image Andrew Brunson has been suffering health problems
Nate Schenkkan, Director for Special Research at Freedom House, called the ruling a "case study in the absurdity of the present Turkish justice system".
Two officials outside the State Department confirmed to NBC News that Pompeo, like Director of National Intelligence Dan Coats, was not aware that National Security Advisor John Bolton directed Press Secretary Sarah Sanders to tweet a White House invitation to Russian President Vladimir Putin last Thursday.
'I believe in and support Turkey's territorial integrity, ' he told the court.
Arrested after the failed Turkish coup attempt in 2016, Brunson was charged with supposed ties to the Kurdistan Workers Party and US -based Muslim cleric Fethullah Gulen, whom Turkey believes inspired much of the turbulence.
Ties between North Atlantic Treaty Organisation allies Turkey and the United States have been strained by the pastor Brunson case. The talks in Manhattan allegedly involved conversations about abducting Fethullah Gulen, the Pennsylvania-based preacher accused of plotting the abortive putsch.
On Wednesday, Brunson was escorted out of prison in the coastal city of Izmir, and left in a convoy of cars.
Erdogan suggested past year that Brunson's case could be connected to an extradition request for Gulen, who lives in Pennsylvania.
"We're working diligently on that case and, frankly, every place an American is held", Pompeo noted, adding that the Brunson case illustrates why advancing religious liberty around the world is so important. Gulen denies the claim.
Pence's announcement of possible sanctions was delivered at a conference on religious freedom in Washington.