"They had them, but they didn't work, because the police turned up and they were shot down and they didn't have the time to get these vests to work. Because had they done so, we would have had major catastrophe," President Beji Caid Essebsi told CNN's Christiane Amanpour.
A third museum attacker remains at large, Essebsi said.
Previously two suspects had been identified -- Yassine Labidi and Saber Khachnaou -- though it wasn't immediately clear if they were the pair killed at the museum by Tunisian security forces.
Essebsi told Amanpour 15 arrests had been made since Wednesday's attack, in which 23 people were killed.
Most of the victims were foreigners
and 19 were tourists who'd been on two cruise ships that docked in Tunis.
Essebsi said he believed the attackers were members of ISIS who had been trained in Libya. This echoed comments made by Tunisia's Security Minister Rafik Chelly on Friday, who said two attackers had been given weapons training at Libyan camps and had been activated from sleeper cells.
In an audio message posted online Thursday
, ISIS claimed responsibility for the attack. The message said the group had targeted "crusaders and apostates" with "automatic weapons and hand grenades" in bloodshed that was "just the start." CNN cannot independently verify the legitimacy of the audio statement.
An uncle of suspect Yassine Labidi, Abeld Malik Labidi, told CNN on Friday that no one he knew had seen any signs of extremism in his 26-year-old nephew.
He said Yassine Labidi was one of the two gunmen killed at the museum.
"It's true that Yassine carried out this terrorist attack, he was killed; his head, his body, we don't have it back," he said.
But, he said, he believed Yassine and other young Tunisians like him were also victims of terrorism -- of the recruiters who paid them money, organized the logistics and took them to places like Syria and Libya to train as fighters.