"We have asked our American counterpart to bring an end to this unlawful [case]," Abdulhamit Gul told journalists in the northern Samsun province.
Atilla, former deputy CEO of state-lender Halkbank, was arrested in the U.S. earlier this year on charges of violating sanctions against Iran. However, he has rejected accusations of evading sanctions.
The minister said it was the duty of Turkish courts to judge if a crime had been committed or not.
"Turkish judges will rule for actions in Turkey," he said.
In his letter to Jeff Sessions sent earlier this week, Gul wrote: "We watch regrettably that this case, which was opened in order to protect the financial interests of the U.S., is proceeding to serve the purpose of completing an unfinished operation of a terrorist organization against Turkey."
He stated that while Turkey was expecting the extradition of members of the Fetullah Terrorist Organization (FETO), the U.S. protects the illegitimate efforts of the terrorist organization.
He said Turkey had asked for the extradition of Huseyin Korkmaz, a fugitive alleged FETO member and former Turkish police supervisor, who fled to the U.S. after being released in February 2016. He is currently testifying in a case against Atilla.
Gul also added that Turkey had asked for the extradition of FETO leader Fetullah Gulen, who has lived in the U.S. since 1999.
Ankara has sought Gulen's extradition in relation to the failed July 15, 2016 coup that left 250 people martyred and nearly 2,200 injured.