News reports two senior administration officials and another person briefed on the matter say certain charges against Brunson will be dropped at a court hearing scheduled for Friday.
According to a report on NBC News, the agreement was reached "recently", and the release is expected to come after a Friday hearing in Turkey's top court leads to the dropping of espionage charges against Brunson.
"We continue to believe Pastor Brunson is innocent, and the hearing on Friday is another opportunity for the Turkish judicial system to free an American citizen", one of the senior administration officials was quoted by NBC as saying.
Late on Wednesday, U.S. Secretary of State Mike Pompeo said releasing Brunson would be the right thing for Turkey to do.
Brunson has been under arrest since December 2016 and under house arrest since July.
USA pastor Andrew Brunson, accompanied by Turkish security officers, arrives at Aliaga Prison and Courthouse complex in a vehicle in Izmir, Turkey October 12, 2018.
The authorities accuse him of having links with the outlawed Kurdistan Workers' Party (PKK) and the Gulenist movement, which Turkey blames for a 2016 failed coup.
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Brunson faces up to 35 years in jail if convicted.
Turkey has good financial and strategic reasons to release the American, analysts say, including the threat of further United States sanctions related to Brunson, and its purchases of Russian weaponry and Iranian oil. Charges against Brunson would be reduced and he would be sentenced to time served or be allowed to serve any remaining term in the United States, the Post said, citing US officials and people close to the case.
"We remain hopeful that with the court proceeding tomorrow that Turkey will see its way clear and free this good man who is guilty of nothing and who has been incarcerated for several years in Turkey unjustly", the vice-president told reporters at a briefing.
Brunson has been accused of backing outlawed Kurdish rebels and the network led by USA -based Muslim cleric Fethullah Gulen.
"We are determined to fight this twisted attitude that has attempted to impose sanctions on our country by using a pastor who has dark ties to terror groups as the pretext", Erdogan said.
The Thursday comments come a day after US Secretary of State Mike Pompeo said he was hopeful that Turkey would release the pastor soon, ending a two-year detention that has shaken relations between the one-time allies.
The row with the United States has exacerbated pressures on Turkey's lira, which has plunged more than 40 percent this year over concerns about Erdogan's influence on monetary policy, relations with Washington and the central bank's ability to rein in double-digit inflation. Brunson has denied the accusation - as has Gulen - and Washington has demanded his immediate release.