WASHINGTON—Several top Democratic members of Congress are asking the Defense Department to investigate whether retired Lt. Gen. Mike Flynn, President Donald Trump’s national security adviser, violated the Constitution when he accepted money from a Russian television network that U.S. intelligence officials say is part of a state-funded media apparatus.

In a letter sent Wednesday to Secretary of Defense Jim Mattis, the lawmakers said that since Mr. Flynn retired from the Army in 2014, he has appeared regularly on Russia Today,...