"He managed to get through all the inspections at Najran and Jeddah airports and travelled on his [the prince's] private jet," it said.
Abu-Bakr al-Qirbi, the Yemeni foreign minister, told The Associated Press news agency that the bomber had claimed he wanted to turn himself in to Saudi officials.
Al-Asiri was in Yemen, al-Qirbi said.
"He claimed that he was going to hand himself over to Saudi authorities and make a statement to his followers to abandon al-Qaeda principles."
"This incident will not change this policy by which we open the door for those who repent"
Prince Nayef bin Abdul Aziz,
Saudi Interior Minister
Mohammed bin Nayef was meeting Ramadan well-wishers when the suicide bomber blew himself up.
It is customary for senior members of the royal family to hold open gatherings during the Muslim month of fasting where citizens can air grievances or offer congratulations.
Prince Nayef bin Abdul Aziz, the interior minister and father of the target of the attack, defended On Saturday the kingdom's efforts to offer rehabilitation to al-Qaeda fighters and their supporters who claim to be repentant.
"The security efforts and strategy that the country is following for reform will not change," he told businessmen ata gathering in Jeddah.
"This incident will not change this policy by which we open the door for those who repent."
The bombing was the first assassination attempt against a member of the royal family in decades and the first significant attack by fighters in Saudi Arabia since 2006.
Al-Qaeda fighters, including those returning from Afghanistan and Iraq, have established sanctuaries in Yemen, particularly in three provinces bordering Saudi Arabia.
Saudi Arabia has waged a campaign against al-Qaeda in the country, killing or capturing most of its leaders after a string of attacks that started in 2003.
Earlier this month, Saudi authorities announced the arrest of 44 people thought to be linked to al-Qaeda and the seizure of explosives, detonators and firearms.
Human Rights Watch has criticised the country for detaining thousands without charge while Amnesty International said the state had committed human rights violations on a "shocking" scale as part of its crackdown.