The unknown individual posted a series of messages on July 29 declaring his intention to massacre audience members at New York's Longacre Theatre, where the legendary boxer is performing his one-man show Undisputed Truth.
Among the suspect's threats was a direct reference to last month's shooting at a midnight screening of The Dark Knight Rises in Aurora, Colorado, for which James Holmes has been charged with murdering 12 people and injuring 58 more.
The Twitter user accused of targeting Tyson wrote: "I might just shoot up this theater in new York I know they leave their exit doors unlocked. Ha now I gotta plan it step by step.
"I wanna kill alot of people...this s**t ain't no joke yo - I'm serious, people are gonna die like aurora."
Authorities have claimed that their investigation is being hampered by a lack of co-operation from Twitter, which has so far refused to provide the individual's contact details.
"We'll cover the theater until we find this guy," police spokesman Paul Browne told The New York Post. "We needed a name and we needed it fast. Twitter said no."
Executives at the website allegedly explained to the NYPD in an email that they are withholding the information because they do not believe "a threat is present, specific and immediate".