Mr Evanoff told reporters: "Letters reportedly written by Osama and seized from the Abbottabad Pakistan compound - where he was killed by US service members - indicate that he was grooming Hamza to replace him as a leader in al-Qaeda.
"Since at least August 2015, Hamza has released audio and video messages on the internet, calling on his followers to launch attacks against the United States, and its western Allies.
"And he has threatened attacks against the United States in revenge for the May 2011 killing of his father."
Osama bin Laden was killed in a US military raid in Pakistan in May 2011.
Mr Evanoff said that since January 2017, the US Department of State has seen Hamza bin Laden as a "specifically designated global terrorist".
He added: "All of Hamza Bin Laden's assets, based in the United States, or in the possession or control of US persons are frozen, and US persons are generally prohibited from engaging in transactions with him."
US Coordinator for Counterterrorism, Nathan Sales, added:
"Today's al-Qaeda is not stagnant - it is rebuilding, and it continues to threaten the United States and our allies.
"In recent years the world's attention understandably has been focused on the ISIS threat.
"Al-Qaeda during this period has been relatively quiet, but that is a strategic pause, not a surrender."
"Make no mistake - al-Qaeda retains both the capability and the intent to hit us."
Al-Qaeda was the group that carried out the 9/11 terrorist attacks in the US.