Terror Intelligence Improvement Act of 2021 This bill establishes a framework to limit the transfer of a firearm or explosive to a person who is or has been under a federal terrorism investigation. Specifically, the bill requires each federal department or agency to provide to the Federal Bureau of Investigation (FBI) information about a person who is or has been under a federal terrorism investigation. Additionally, it requires the FBI and its Joint Terrorism Task Forces to immediately be notified of a request to transfer a firearm or explosive to a person who is, or was within the previous 10 years, under a federal terrorism investigation (i.e., a suspected terrorist). Further, the Department of Justice must establish a process to delay and prevent the transfer of a firearm or explosive to a suspected terrorist.