A Yemeni government official, speaking on condition of anonymity, said officials are waiting for DNA confirmation.
The killing of al-Shihri and other AQAP leaders "is leading to the gradual dismantlement of the group," CNN National Security Analyst Peter Bergen said.
White House counterterrorism adviser John Brennan in April described AQAP as "very, very dangerous" and "the most active operational franchise" of al Qaeda.
FBI Director Robert Mueller told Congress in May that al Qaeda and its affiliates, "especially al Qaeda in the Arabian Peninsula, currently represent the top counterterrorism threat to the nation."
The group was behind the so-called underwear bomb attempt on a Christmas Day U.S.-bound international flight in 2009 and an effort to smuggle bombs in printer cartridges onto U.S.-bound cargo planes in 2010.
Al-Shihri, who was once held by the United States at Guantanamo Bay in Cuba, was killed Monday in an operation in Hadramawt Valley, state-run news agency SABA reported.
Six other terrorists were killed as well, the military said in its statement, posted on an official website.
While the Yemeni military said al-Shihri was killed in "an operation by the armed forces," three local security officials told CNN that a U.S. drone conducted the strike.
The United States generally does not comment on reported drone strikes.
"For weeks we have known his destination and were waiting for the right time to conduct this major operation. The attack makes al Qaeda a handicapped network," a senior Defense Ministry official told CNN on condition of anonymity because he is not authorized to talk to media.
Officials: Al Qaeda affiliate surrounded at last Yemen stronghold
In November 2007, al-Shihri, a Saudi national, was transferred from Guantanamo Bay to Saudi custody to undergo a program designed to lead people away from terrorism.
Despite travel restrictions requiring him to stay in Saudi Arabia, he left for Yemen, where he joined another former Guantanamo detainee to assume leadership of AQAP, the U.S. House Armed Services Committee said in a document about detainees who resume terrorism.
The U.S. State Department notes that he was publicly identified in January 2009 as the deputy leader of AQAP.
"As deputy of AQAP, al-Shihri helps carry out terrorist acts by generating targets, recruiting new members, assisting with training and attack planning, and tasking others in the preparation of attacks," the State Department said.
He is believed to have played a key operational role within AQAP, including the planning of attacks inside Yemen and a failed attack to the kill the head of Saudi counterterrorism in 2009, Cruickshank said.
Al-Shihri "was the most important Saudi figure within the group. He was key to AQAP's recruitment of Saudis and their fundraising in the kingdom," Cruickshank said.
In December 2009, Yemen said it thought al-Shihri was at the site of an aerial assault against senior al Qaeda operatives. But later reports said he had apparently escaped the attack.