"We will withdraw our operation there to save the life of the hostage, our driver," Faisal al-Nahait, owner of Faisal al-Nahait Transport Company, where Muhammad al-Gharabawi has worked for eight years, told Aljazeera television.
On 13 July, a purported Iraqi group said it would behead al-Gharabawi, who it seized a week earlier, unless the Saudi company pulled out of Iraq within 72 hours.
Al-Gharabawi, in his 50s, was abducted while delivering petrol products to US forces in Iraq by a group claiming to be the the "legitimate Iraqi resistance", Aljazeera reported last week.
Al-Nahait, had earlier claimed that unidentified callers had telephoned him several times since Wednesday demanding money.
He urged Egyptian authorities to help secure the driver's freedom.