The Justice Department will seek to indict former FBI director James B. Comey for allegedly giving false testimony to Congress about his role investigating efforts to influence the 2016 presidential election, according to people familiar with the matter, days after President Donald Trump demanded prosecutors use the criminal justice system to punish his political opponents.

The investigation centers on testimony Mr. Comey gave before the Senate Judiciary Committee on Sept. 30, 2020, about the FBI’s missteps in the “Crossfire Hurricane” probe, which had delved into possible but ultimately unproven collaboration between Russia and Trump’s 2016 presidential campaign.

Under the statute of limitations, the deadline to seek an indictment against Mr. Comey is Tuesday, five years from the date o

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