In social media posts over the past few months, Chase Bliss Colasurdo of Kent, Washington, also threatened to kill one of the president's sons, Donald Trump Jr., and to "start bombing synagogues," court documents alleged.
Colasurado has been charged with two counts of making interstate threats.
The FBI said it began its investigation in mid-March based on a tip from what authorities describe as a "concerned citizen."
Authorities discovered that on Feb. 26, Colasurdo allegedly sent an email to five different media outlets stating, "I'm going to personally execute [Kushner] for his countless treasonous crimes," according to charging documents.
The very next day he posted on his publicly accessible Instagram account a photo of himself holding what appeared to be a gun aimed at a picture of Kushner, charging documents allege.
"This sneaky mossad agent has never been so serious about something as he is about putting a bullet in [Kushner's] brain. It must be done," charging documents said the suspect stated, referring to Mossad, the Israeli intelligence agency.
Later that day, court documents said, he posted another photo of himself and a handgun bragging about making death threats against Kushner and not getting arrested.
In March, according to court documents, during his FBI interview Colasurdo made "numerous anti-Semitic comments expressing hostility against Jews."
Court documents also allege that Colasurdo attempted to purchase a firearm in April and was denied due to an alert entered about him by the Secret Service.
After executing a search warrant, authorities found that he had among other items, firearm holsters, 345 rounds of ammunition, a Nazi flag with a swastika and a framed portrait of Adolf Hitler along with some of his writings.
The government says that Colasurdo has a history of mental illness and two arrests for assault.
He is due in court on May 6 for a detention hearing.
ABC News' Mike Levine contributed to this report.