His captors have threatened to kill him on Friday, but it's not clear at what time the deadline expires.
While Saudi security forces continue their search for Johnson, the US State Department has again told its citizens remaining in Saudi Arabia to leave the country.
The department on Thursday reissued a previous travel warning "to remind American citizens of the continuing serious threat to their safety while in Saudi Arabia".
It said any Americans planning travel to Saudi Arabia should cancel their plans. The US government has already evacuated dependents and non-emergency workers at its embassy and consulates in the country.
The State Department said it reissued the warning in the wake of last week's series of attacks in Saudi Arabia that targeted Americans.
Two were shot dead, and Johnson, 49, an engineer for defence contractor Lockheed Martin, was kidnapped.
Three days later, Johnson appeared on a video on the internet in which his captors identified themselves as an al-Qaida group.
They threatened to kill him on Friday unless the Saudi government releases all its al-Qaida prisoners and all Westerners leave the Arabian peninsula.
"The US government continues to receive credible information indicating that extremists are planning further attacks against US and Western interests," the warning said.
"Credible information indicates that terrorists continue to target residential compounds in Saudi Arabia, particularly in the Riyadh area, but also compounds throughout the country.
"Recent incidents indicate that American citizens residing in private residences are also being specifically targeted," it added.
Wife's TV plea
Menawhile Johnson's Thai wife has pleaded for his release on Arab satellite television as the deadline neared.
"It really hurt me so bad to see him on TV," she told Al-Arabiya on Friday.
"What I can do for him, I will," she said, speaking in broken English.
"I want him to come back to me ... He didn't do anything wrong," she said, as she started crying and was barely able to speak.
The wife, identified as Thanom by a Saudi newspaper, said she was worried about Johnson because he was taking medication for diabetes.
She also told the Dubai-based channel that he has problems with his ankle and was unable to walk sometimes.
"I have nobody in Riyadh. I only friends in the US and Bangkok to call and to pray" for her husband.
"That's why I be here by myself waiting for him to come back and see me," she added.