The Los Angeles Times first reportedin January that al-Awlaki, believed to be in hiding in Yemen, had been added to a list maintained by the US military's Joint Special Operations Command that authorises forces to capture or kill him.
Al-Awlaki has been linked to Major Nidal Hasan, a US army psychiatrist accused of killing 13 people, and Umar Farouk Abdulmutallab, a Nigerian student accused of trying to blow up a Detroit-bound flight on Christmas Day.
The US believes he plays a major role in setting strategy for al-Qaeda in the Arabian Peninsula (AQAP).
In April, the New York Times also reportedthat the administration of Barack Obama had authorized al-Awlaki's inclusion on both the military and Central Intelligence Agency's target lists.
Al-Awlaki is likely the first American citizen to be added to either list, both newspapers reported.
The two rights groups said they were retained in June by al-Awlaki's father, Nasser, to bring a lawsuit over the alleged kill orders.
But on July 16, the US treasury department addedal-Awlaki to a list of "specially designated global terrorists," freezing his assets and outlawing any dealings with him.
The designation forced the rights groups to apply for a special license to file suit on behalf of Nasser al-Awlaki.
The "ability to file a lawsuit as quickly as possible may spell the difference between life and death for Anwar al-[Awlaki]," the groups wrote in their Tuesday court filingto force the treasury department to grant them the license.
"While we appreciate [the government's] quick response to our lawsuit, we continue to believe that [the treasury department's] regulations are unconstitutional because they require lawyers who are providing uncompensated legal representation to seek the government’s permission before challenging the constitutionality of the government’s conduct," the groups' statement said.
According to the treasury department's statement in July, al-Awlaki is a key leader of AQAP and has pledged an oath of allegiance to its emir, Nasir al-Wuhayshi.
Al-Awlaki was imprisoned in Yemen in 2006 on charges of kidnapping for ransom and being involved in an al-Qaeda plot to kidnap a US official but was released from jail in December 2007 and subsequently went into hiding in Yemen, according to the treasury department.
On Tuesday, the White House pushed back strongly against the accusations that the government was trampling on civil liberties.
"Let's be clear about Anwar al-Awlaki," Robert Gibbs, a White House spokesman, said.
Awlaki "is someone who has sworn allegiance to Al-Qaeda in the Arabian Peninsula, is a regional commander for that group in Yemen, has and continues to direct attacks there and, as we know, against innocent men, women and children in this country," he told reporters.
"And this president will take the steps necessary to keep our country safe from thugs like him."