CAIRO, Nov 26 (Reuters) - Police in Cairo are investigating the disappearance of an Egyptian-born German author and critic of Islam, security sources said on Tuesday, months after a cleric declared him an infidel and called for his death.

Hamed Abdel-Samad, author of "The Downfall of the Islamic World: A Prognosis", went missing in recent days, according to a missing person's report filed by his brother, police sources said. An Egyptian newspaper report said he had been abducted.

The German ambassador in Cairo has asked Egypt's deputy prime minister, Ziad Bahaa El-Din, to do everything to secure his "personal freedom and physical well-being", a German Foreign Ministry spokesman said on Monday.

He added Abdel-Samad, 41, had been in contact with the German embassy in Cairo regarding his personal safety, without giving further details. His whereabouts were unknown and the German authorities were working hard to solve the case, he said.

In a television show broadcast in June and posted on YouTube, an Egyptian cleric, Mahmoud Shaaban, called for the killing of the author, who frequently appears in German media as a speaker on Islamic affairs, saying he was an infidel.

Hardline Islamists were oppressed by the state for years but allowed to speak more freely after the downfall of veteran autocrat Hosni Mubarak in 2011. Earlier this year, Shaaban, a vocal supporter of deposed Islamist President Mohamed Mursi, said liberal politician Mohamed ElBaradei also deserved to die.

The Egyptian security forces have been cracking down on Islamists since the Muslim Brotherhood's Mursi was deposed by the army in July following mass protests against his rule.

In the 1990s, hardliners attacked several prominent Egyptian cultural figures. The late Nobel prize winning novelist Naguib Mahfouz survived a knife attack in 1994. (Reporting by Shadia Nasralla and Asma Alsharif; Editing by Tom Perry and Alison Williams)