Mr Johnson's wife pleaded for his release on Arabic TV
Al-Qaeda militants are reported to have beheaded an American engineer they have been holding hostage in Saudi Arabia since last week.
Arabic television network al-Arabiya said Paul Johnson had been executed, while an Islamist website showed photographs of the apparent beheading.
Al-Qaeda had given the Saudi government a Friday deadline to free jailed militants, or it would kill Mr Johnson.
Mr Johnson, 49, was seized in the Saudi capital, Riyadh, last Saturday.