The men were charged with spreading false information harmful to national security, two security sources familiar with the incident said, on condition of anonymity.
Egypt’s interior ministry could not be reached for comment after several attempts to contact it on Wednesday.
Shafik announced last month his intention to run against President Abdel Fattah El-Sisi in Egypt’s upcoming presidential election early next year. No date has been set for the vote.
The former premier and ex-air force commander is seen by Sissi’s critics as the most serious potential challenger to the incumbent. Those arrested were all members of Shafik’s Egyptian National Movement party.
One of them, Hany Fouad, was arrested at his family’s home on the outskirts of Cairo, his brother Wael Fouad said.
“Hany is very close to (Shafik),” he said.
Security forces entered the family home around 3 a.m. local time (0100 GMT) and searched the house, confiscating two mobile phones and giving no reason for the arrest, he said.
Sissi has yet to announce that he will seek a second term but is widely expected to do so.
Shafik made his announcement from the United Arab Emirates where he has lived in exile since 2012, when he fled Egypt after a narrow electoral defeat to former president Mohamed Mursi of the Muslim Brotherhood, who was removed from office in a military takeover led by Sissi in 2013.
Shafik returned to Cairo ten days ago and said he was still deciding whether to run. He has not made any public appearances since.