Prince Saud al-Faisal, the Saudi foreign minister, said on Tuesday that Riyadh had contacted international organisations, including the Red Cross, about the man.
In Israel, a spokeswoman for the country's Prison Service said: "He is actually not a prisoner but an illegal infiltrator into Israel. He has been with us for 11 months."
The spokeswoman named him as Abdul-Rahman al-Atawi and said he was seeking asylum through the UN. The Red Cross was also believed to be involved in the case.
"What I know about him is that ... the UN is attempting to place him in a country for absorption and not return him to Saudi [Arabia]. I understand attempts have been made to get him asylum in Turkey, Jordan, Sweden and Finland without any success," she said.
The spokeswoman said the man was believed to be the only Saudi held in Israel. The two countries have no diplomatic relations.