The attack left 23 people dead, including 20 foreign tourists.
It alleged that four were directly linked to the attack and five had “ties to the cell”.
The army will also be deployed to major cities, the presidency added.
In another development, Islamic State (ISIS) said in an audio message that it was behind the attack.
It said the attack was carried out by “two knights of the caliphate” and named them as Abu-Zakariya al-Tunisi and Abu-Anas al-Tunisi.
A statement described the attack as a “blessed invasion of one of the dens of infidels and vice in Muslim Tunisia”.
The statement was published by Twitter accounts known to be reliable sources of ISIS propaganda.
One of two gunmen involved in the Bardo Museum attack, named by Tunisian officials as Yassine Laabidi, was reportedly known to the authorities.
Yassine Laabidi and his accomplice, named as Hatem Khachnaoui, were killed as security forces stormed the museum.
It was not immediately clear how the identities of the gunmen corresponded to the names given by ISIS. Jihadist groups, including ISIS, often give their fighters noms de guerre.
Tourists from Japan, Colombia, the UK and other European countries were killed in the attack and more than 40 people were injured.
The suspects arrested on March 19 were not identified and no further details of their alleged involvement were given.
In a statement, the presidency said Tunisia was facing “exceptional circumstances”, adding that “terrorist operations have now moved from the mountains to the cities”.
“After a meeting with the armed forces, the president has decided large cities will be secured by the army,” the statement added.
On March 19, two Spanish tourists and a Tunisian museum worker were found at the museum after having hidden there overnight believing the attack might not have been over, police said.
Two cruise companies – MSC Cruises and Costa Cruises – said they were suspending stopovers in Tunis.
MSC said in a statement that at least nine of those killed had been passengers on its MSC Splendida cruise ship which was docked in Tunis. It said another 12 of its passengers were injured and six were still unaccounted for.
Costa Cruises said that three passengers from the Costa Fascinosa had died. Eight others were injured and two were unaccounted for, company CEO Michael Thamm said in a statement.
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