The row over Brunson escalated last week when US Vice President Mike Pence, like the pastor an evangelical Christian, said Turkey would face "significant sanctions" if this "innocent man of faith" was not freed.
The sanctions, announced Wednesday, allow the US government to target individuals and companies involved in corruption or human-rights abuses, which in this case include Turkey's justice and interior ministers.
Speaking to Turkish journalists after his meeting with Pompeo, Cavusoglu also described their discussion as "extremely constructive" and said the two would continue to work toward resolving disputes.
Turkish Foreign Minister Mevlut Cavusoglu said he and Pompeo agreed to move forward with diplomatic negotiations.
"The Turks were on notice that.it was time for Pastor Brunson to be returned", Pompeo told reporters in Singapore.
Brunson is charged with supporting the group Ankara blames for orchestrating an attempted coup in 2016.
Turkish Finance Minister Berat Albayrak, who is also Erdogan's son-in-law, said the sanctions were unacceptable and would have a limited impact on the Turkish economy.
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The Turkish lira, which has been in decline against the dollar for most of 2018, fell to a record low of five to the dollar on Wednesday, after dropping 1.7 per cent following the sanctions announcement.
"However, Turkey always wishes to resolve these issues via diplomacy, dialogue and mutual understanding", Cavusoglu told reporters after meeting his USA counterpart Mike Pompeo in Singapore.
Brunson, 50, is being tried on espionage and terror-related charges, which he and the USA government vehemently deny. He was recently moved from prison to house arrest. They have been at odds over numerous matters, including military activity in northern Syria and Turkey's plans to purchase an advanced air defense system from Russian Federation.
Gulen, who has lived in self-imposed exile in the USA since 1999, has denied any role in the 2016 coup attempt.
Turkey has been trying to have Gulen extradited from the United States for two years.