Al-Badawi was later convicted of plotting, preparing and helping carry out the attack on the USS Cole.
Abd al Rahim al-Nashiri, another al Qaeda militant, has been in United States custody since 2002 in connection with the Cole attack.
There is a five-million dollar reward for information leading to his arrest.
The expanded operations included intensified overnight airstrikes and ground military raids against the al-Qaida hideouts in the mountainous areas of al-Bayda and southeastern province of Shabwa.
US forces conducted a precision airstrike targeting al-Badawi on January 1 in the Marib governorate, said Capt. Bill Urban, a Central Command spokesperson.
USA forces carried out an airstrike in Yemen on New Year's Day targeting an Al Qaeda operative suspected of having involvement in the deadly 2000 attack on the USS Cole, a military official confirmed Friday.
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USA intelligence indicates al-Badawi was killed Tuesday in an airstrike in Ma'rib Governorate, CNN reported, citing an anonymous Trump administration official. No collateral damage is now known.
Al-Badawi's death in Tuesday's airstrike was also confirmed by Voice of America, citing a defense official.
In October 2000, 17 American sailors from the Cole, a guided-missile destroyer, died in a suicide bomb attack at Aden harbor.
"Jamal al-Badawi was indicted by a federal grand jury in 2003, charged with 50 counts of various terrorism offenses, including murder of US nationals and murder of USA military personnel", the statement said. He was recaptured in March 2004 and sentenced to death, but almost two years later he managed to escape a second time.
Al-Badawi had been held in custody by Yemeni authorities, but escaped in April 2003. He managed to escape a second time in February 2006, according to the Federal Bureau of Investigation.