"The kidnappers called me a few minutes ago saying that they have decided to release him tomorrow morning," said Faisal bin Ali al-Nahait, whose company carries the same name.
He told reporters an unidentified caller telephoned from Iraq to tell him Egyptian truck driver Muhammad al-Gharabawi "would be freed tomorrow".
"They will call me tomorrow [Sunday] to inform me of the place," where al-Gharabawi will be set free, Nuhait added.
Al-Nahait had announced on Wednesday he would comply with the captors' demands in order to save al-Gharabawi's life.
The company confirmed on Friday it had ended all its operations in Iraq and withdrawn its vehicles.
Al-Nahait provides trucks and drivers to the Saudi Arabian al-Jari transport company, which is licensed to ferry oil shipments to Iraq through the Arar border crossing.
A group holding Gharabawi issued a statement late on Thursday, giving the Saudi employers an extra 48 hours to leave the country and spare his life.