The prosecutor accuses Mr Fuller, former vice-chair of the US National Intelligence Council, of having links to Muslim cleric Fethullah Gulen.
Mr Gulen, who lives in the US, is blamed by Turkey for last year’s failed coup attempt. He denies the claim.
The warrant also accuses Mr Fuller of “attempting to overthrow” Turkey’s government, the Hurriyet daily reports.
More than 50,000 people have been arrested in Turkey pending trial over links to Mr Gulen.
Another 150,000 people have been sacked or suspended from jobs in the public and private sectors since the coup attempt on 15 July 2016.
Turkish-US relations have been strained in recent months.
Reza Zarrab is a key witness in the criminal trial of a Turkish banker whom he allegedly worked with to help Iran launder money.
President Erdogan has denied that Turkey breached US sanctions on Iran.
Just days before that row, Turkey denied reports that former White House aide Michael Flynn was part of a plot to help forcibly remove Mr Gulen to Turkey in exchange for millions of dollars as “ludicrous”.
In October, Washington and Ankara were involved in a consular row, mutually scaling back visa services.