Suleiman, who was intelligence chief under former President Hosni Mubarak, submitted his candidacy for the presidency Sunday shortly before the filing deadline after collecting 72,000 signatures.
"And indeed as soon as my nomination for the presidency was announced, I received on my personal mobile and through some people close to me death threats and messages saying we will take revenge -- from members of the Muslim Brotherhood and other Islamist groups," Suleiman told the Egyptian daily al-Akhbar Monday.
Suleiman said he decided to submit his presidential candidacy independently. He said the country's ruling military council was involved in his decision.
"The [ruling] Supreme Council has no relation, neither negatively or positively, with my decision to join the race for the presidency," he told the newspaper.
The Israeli daily Yedioth Ahronoth said two other presidential candidates are Amr Moussa. former Arab League secretary-general who served for 10 years as foreign minister under Mubarak, and Khayret al-Shater of the Muslim Brotherhood's Freedom and Justice Party.
The Egyptian Web site Ahram Online said al-Shater's candidacy was in doubt because served a jail sentence on charges of terrorism and money laundering. He was released from jail in March of 2011.