The case revolves around the suspicion that the men’s association with members of Al Qaeda and other militant groups in the course of their reporting has led to their classification as terrorists.

Image Ahmad Muaffaq Zaidan in 2004. Mr. Zaidan said that documents from the National Security Agency leaked by Edward J. Snowden suggest that he is on a list of people marked for death by the United States government. Credit Anjum Naveed/Associated Press

Mr. Abdul Kareem said he believed that he was on the list because he had nearly been killed in airstrikes five times in the last year, at least one of them from a drone, according to court papers. According to Clive Stafford Smith, the Founder of Reprieve, Mr. Abdul Kareem was also informed by a Turkish intelligence official that the United States was seeking to kill him.

Mr. Zaidan, who has both Pakistani and Syrian citizenship, thinks he is on the list based on documents from the National Security Agency leaked by Edward J. Snowden and published by The Intercept.

The documents appear to be slides from a presentation about a technology that uses metadata from cellphones to identify couriers for Al Qaeda. One slide contained a picture of Mr. Zaidan, alleging that he is a member of Al Qaeda and the Muslim Brotherhood, in addition to being an employee of Al Jazeera. It also showed that he had an identification number in the Terrorist Identities Datamart Environment, or TIDE, the lowest level of the government’s terrorism watch list.

After the information was published, Mr. Zaidan, fearing for his life, fled Pakistan for Qatar, where Al Jazeera is based, according to the lawsuit.

The case, filed in the United States District Court for the District of Columbia, names 12 defendants all believed to be involved in the United States’ covert drone program. They include Mr. Trump, the heads of the Central Intelligence Agency and Department of Homeland Security, and the departments of Justice and Defense.

The government has 60 days to respond, and could argue that the court lacks jurisdiction to rule on such matters.