The notes are the first time the face of regime leader Bashar al Assad will appear on Syrian currency since he took office 17 years ago.
A State Department official declined to comment to the magazine on the private discussion, but insisted that the USA remained "committed to the Geneva peace process" and supported a "credible political process that can resolve the question of Syria's future".
"Ultimately, this process, in our view, will lead to a resolution of Assad's status", the official told Foreign Policy".
Tillerson's latest comments contradict earlier assurances that the USA would demand that Assad cede power after the regime allegedly used chemical weapons against civilians in April.More news: India ready for Sri Lanka
"The Syrian people should determine their country's political future through a political process", the official added.
The Russian Defence Ministry called it "a cynical violation of the sovereignty of the Syrian Arab Republic".
However, rebel leaders accused the army of nearly immediately carrying out barrel bomb attacks in opposition-held areas in the city of Daraa and the town of Naima and the Daraa countryside.More news: Dallas shooter in police custody
The bill featuring Assad's portrait was put into circulation "in Damascus and a number of the provinces".
He also signaled that USA military attacks on Syrian government forces in recent months were of tactical nature and not meant to weaken the government or strengthen the opposition forces.
Foreign Policy reported, citing three unnamed diplomatic sources familiar with the exchange, that Tillerson made the remarks to U.N. Secretary General António Guterres last week in a private State Department meeting.
Tillerson's remarks to the United Nations leader reflect the political reality that Assad will likely be the victor in Syria's civil war, now grinding along in its sixth year.More news: Phil Simmons applies for India's head coach job
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