"Right now there does not appear to be any criminality. We're just waiting for the M.E. (medical examiner) to determine a cause of death," said Officer Sophia Mason of the New York Police Department.
The man apparently collapsed due to a medical issue, NY Daily News reports.
Police responded less than an hour after polls opened in New York following a presidential campaign in which Russia was a leading issue.
The Russian consulate is less than two miles (three km) from Trump Tower, the home and campaign headquarters of Republican candidate Donald Trump. Democrat Hillary Clinton voted in suburban Chappaqua, New York, shortly after the call came in.
Upon arrival at the consulate, officers found the man unconscious, and paramedics later pronounced him dead at the scene, police said in a statement.
Police were withholding identification of the man pending family notification. Russian consulate officials were unavailable for comment.