The U.S. request was made recently, the report said.
The Egypt Independent had a similar report, saying it was told by Mamish the Navy refused a United States request to "strike" the Iranian ship.
Mamish said the authority is notified 24 hours in advance of which ships will pass through the canal and the cargo they are carrying. The information is verified before the ships are permitted to pass, he said.
"My main goal since I was appointed chairman of the Suez Canal Authority [Aug. 12] is to secure all ships passing through the waterway," he said.
The Suez Canal connects the Mediterranean Sea with the Red Sea.
In February of 2011, two Iranian navy vessels were permitted to sail through the canal and were later spotted at the Syrian port of Latakia, Haaretz reported at the time.