An assault in a town in the North Sinai last week killed more than 300 worshipers at a mosque, in the latest indication that the four-year-old militant insurgency centered there threatens Egyptian civilians as well as security forces.

“A change of blood might be what is called for,” Mr. Shafik said in the first video statement, citing the deep downturn in the Egyptian economy, the deterioration of public services and the accumulation of public debt.

The Egyptian government might find other ways to thwart a Shafik candidacy. When he left Egypt after losing the 2012 presidential race, prosecutors were investigating corruption charges against him related to his tenure as the minister of civil aviation, a post he held for a decade beginning in 2001. Those charges might be revived to deter him from returning to Egypt or competing in the race.

If he is able to return and run, Mr. Shafik’s military record and well-known name may make him a strong candidate. He fought in Egypt’s two wars with Israel, in 1967 and 1973, and rose to command the Air Force. As minister of civil aviation, he oversaw the successful modernization and development of both the national airline and the Cairo airport.

He developed strong connections among the business elite, and he was widely discussed as a possible successor to former President Hosni Mubarak. In a bid to bolster his support during his last days in power in 2011, Mr. Mubarak had promoted Mr. Shafik to prime minister in the hope that he could resolve the crisis brought about by a popular uprising.

Mr. Shafik lost the presidential election the next year by only about 2 percent of the vote. The winner was Mohamed Morsi of the Muslim Brotherhood, whom Mr. Sisi removed from office in 2013.

Although Mr. Shafik campaigned against Mr. Morsi in 2012 as a defender of law-and-order and an admirer of Mr. Mubarak, his video on Wednesday suggested that this time he intended to run as a champion of democracy against Mr. Sisi.