The father made the statement about his son being a terrorist to New Jersey police in 2014, when Mr. Rahami was arrested after a domestic dispute and accused of stabbing his brother.
The information was passed to the Joint Terrorism Task Force led by the Federal Bureau of Investigation in Newark. Officers opened what is known as an assessment, the most basic of F.B.I. investigations, and interviewed the father, who then recanted.
An official, when asked about the inquiry, said the father made the comment out of anger at his son.
It is not clear if officers interviewed Ahmad Rahami.