The lawsuit was first reported by Courthouse News Service.
The September drone killing of al-Awlaki in Yemen drew harsh criticism from civil rights activists, who argued that the terrorist leader was not afforded due process and a fair trial.
Republican presidential candidate Ron Paul said the strike was not "a good way to deal with our problems."
"He was born here, al-Awlaki was born here, he is an American citizen. He was never tried or charged for any crimes," Paul said. "No one knows if he killed anybody. We know he might have been associated with the underwear bomber. But if the American people accept this blindly and casually that we now have an accepted practice of the president assassinating people who he thinks are bad guys, I think it's sad."
But the White House defended the decision as necessary to protect American security.
The Times says in its complaint that the government should turn over documents explaining how and under what circumstances such a determination is made.
"Given the questions surrounding the legality of the practice under both U.S. and international law, notable legal scholars, human-rights activists and current and former government officials have called for the government to disclose its legal analysis justifying the use of targeted lethal force, especially as it applies to American citizens," the paper wrote in its complaint.
In Feburary, Newsweek reported that the Justice Department had provided the CIA with legal guidance about the legality of targeted-killing operations.