'If the Saudi government wants Paul Marshall Johnson to be released, it must release the mujahedeen held in the prisons in Hair, Al-Ruwais and Allecha within 72 hours. Otherwise, we will execute him to avenge our Muslim brothers whose blood has been spilled in several parts' of the world, the statement said.
The statement's authenticity was not confirmed although it was signed 'Al-Qaeda in the Arabian Peninsula'.
The Islamic Web site showed a videotape of Johnson wearing a blindfold.
Johnson, an aeronautics engineer working for US defence contractor Lockheed Martin, went missing on 12 June.
On the same day, Crown Prince Abdullah warned that Riyadh's patience with terrorists was wearing thin. He warned there could be no compromise in defending Islam and Saudi Arabia and protecting foreigners in the Kingdom and urged scholars and clerics to do more to fight extremism.
Prince Abdullah predicted the next few days would bring good news in the fight against terrorism. His comments were broadcast live on Saudi television.
'As you know, these despicable incidents were being committed by the sons of this country. As a brother and citizen I must say I am appalled and sickened by what happened,' he said.
He also vowed the Kingdom 'is capable of doing more than it has done. For the sake of mercy toward the errant group, we maintained patience and refrained from doing certain things that I cannot disclose now. We have been patient all this time... but our patience has limits.'