Under Spanish asylum rules, the ministry has 72 hours to make a decision, and the petitioner has a right of appeal.
Omar, 28, will remain at the airport while his request for asylum is processed, an interior ministry spokesman said. If it is accepted, there is a period of two months for the final decision.
Authorities in Britain had turned down a request in April from Omar for a settlement visa.
At the time he said he wanted to live in England with his new British wife, Zaina Alsabah bin Laden, 52, formerly known as Jane Felix-Browne.
Zaina is now in Madrid. She told local media that her husband has asked for asylum in Spain because "he doesn't feel safe in Egypt or in any other country in that area".
The British embassy in Cairo said it had based its decision on fears that his presence in the country would cause "considerable public concern".
Omar is the fourth of 11 children born to his father's first wife, and he is one of 19 children Osama bin Laden has fathered.
In a television interview broadcast in January, Omar urged his father to give up violence.
"I try and say to my father: 'Try to find another way to help or find your goal. This bomb, this weapons, it's not good to use it for anybody,'" he said.
He also said he had not spoken to his father since 2000, when he walked away from an al-Qaeda training camp in Afghanistan with Osama's blessings, and does not know where the al-Qaeda leader is.
Osama bin Laden has claimed responsibility for the September 11, 2001 attacks in the United States that killed more than 3,000 people.