WASHINGTON—Mike Flynn, President Donald Trump’s former national security adviser, has told the Federal Bureau of Investigation and congressional officials investigating the Trump campaign’s potential ties to Russia that he is willing to be interviewed in exchange for a grant of immunity from prosecution, according to officials with knowledge of the matter. As an adviser to Mr. Trump’s presidential campaign, and later one of Mr. Trump’s top aides in the White House, Mr. Flynn was privy to some of the most sensitive foreign-policy deliberations of the new administration and was directly involved in discussions about the possible lifting of sanctions on...