It is not clear whether the father's offer had been made with al-Awlaki's consent.
The US administration reportedly approved the killing of al-Awlaki after US intelligence agencies concluded he was now directly involved in plots against the US, and not just publicly encouraging such attacks.
But Yemeni authorities said on Saturday that they had not received any evidence from the US to support allegations that the US-born al-Awlaki is recruiting for an al-Qaeda offshoot in Yemen.
"Anwar al-Awlaki has always been looked at as a preacher rather than a terrorist and shouldn't be considered as a terrorist unless the Americans have evidence that he has been involved in terrorism," Abu Bakr al-Qirbi, the Yemeni foreign minister, said.
His announcement came after a powerful Yemeni tribe threatened to use violence against anyone trying to harm al-Awlaki.
In an official statement published on Saturday after a meeting of tribal leaders, the Al-Awalik tribe, which is active in the Abyan and Shabwa regions that are important al-Qaeda strongholds, said it would "not remain with arms crossed if a hair of Anwar al-Awlaki is touched, or if anyone plots or spies against him".
"Whoever risks denouncing our son (Awlaki) will be the target of Al-Awalik weapons," the statement said, and gave warning "against co-operating with the Americans" in the capture or killing of al-Awlaki.
Al-Awlaki rose to prominence last year after it emerged he had had prolonged communications with Major Nidal Hasan, a US army psychiatrist accused of opening fire on colleagues at Fort Hood, Texas, killing 13 people.
He is also accused of having had ties to the September 11, 2001, hijackers, and to Umar Farouk Abdulmutallab, the Nigerian student accused of trying to blow up a Detroit-bound flight with explosives on December 25.
Yemen, the poorest country in the Arab world, is beset by serious political and administrative problems.
In addition to the conflict with the regional branch of al-Qaeda, Yemen's weak central government has struggled to contain separatists in the south and Houthi fighters in the north.
The government and the Houthis reached a ceasefire agreement in February. But the separatist problems in the south show no sign of a resolution.