The group, which has hundreds of Tunisians among its ranks, threatened more attacks, saying: "What you have seen is only the start."
Authorities say as many as 3,000 Tunisians have gone to Iraq, Syria and Libya to fight in jihadist ranks, raising fears of battle-hardened militants returning home to plot attacks.
Around 200 demonstrators gathered Thursday outside the museum where the attack killed 20 foreigners and at least one Tunisian, chanting "Terrorism Out" and laying flowers at the entrance.
The president's office said security forces arrested "four people directly linked to the (terrorist) operation and five suspected of having ties to the cell".
And a presidential source said soldiers are to be deployed in major cities following the assault, while insisting "we are not under siege".
As international outrage grew over Tunisia's worst post-revolution attack, President Beji Caid Essebsi said his country would not be cowed by extremism.
"The process of implementing a democratic system is underway, well anchored," he told France's TF1 television. "We will never move backwards."
The leader of the Islamist opposition party Ennahda, Rached Ghannouchi, said he was convinced that "the Tunisian people will stay united in the face of barbarity".
- Appeals for unity -
The media also called for solidarity, with newspaper La Presse appealing for "total unity and a sense of responsibility shared by all".
Panic had broken out as gunmen in military uniforms opened fire at visitors as they got off a bus and then chased them inside the museum.
The dead included three Japanese, two Spaniards, a Colombian, an Australian, a British woman, a Belgian woman, two French, a Pole and an Italian, Health Minister Said Aidi said.
Dozens more people were wounded in the assault, in a massive blow to Tunisia's heavily tourism-dependent economy.
At least two major cruise ship operators suspended stopovers in Tunis following the attack.
After cowering in fear in the museum during the night, two Spanish tourists were discovered alive and well, officials said.
The government, in a show of defiance, said the National Bardo Museum would reopen early next week.
But in a sign of rising tensions, police encircled the national radio headquarters Thursday as a "precautionary measure" following "terrorist threats", said its head, Abderrazak Tabib.
Prime Minister Habib Essid named the two gunmen killed by security forces at the museum as Yassine Abidi and Hatem Khachnaoui. He said Abidi was known to the police.
The museum assailants were "probably" Tunisian, the interior ministry added.
President Essebsi said the gunmen had explosives on them and he praised the security forces for preventing further bloodshed.
"We found terrible explosives on those people that they did not have time" to use, he told TF1.
Tunisia has seen an upsurge in Islamist extremism since the 2011 revolution that ousted longtime strongman Zine El Abidine Ben Ali and sparked the Arab Spring uprisings around the region.
Nine of the slain tourists were from the MSC Splendida cruise ship, whose owners said a special psychology unit had been set up for passengers.
MSC Cruises and Italian operator Costa Crociere said they would divert cruise ships which had been due to berth in Tunis.
Italy's Defence Minister Roberta Pinotti said her country will increase its military presence in the central Mediterranean, "following a worsening of the terrorist threat, dramatically demonstrated by yesterday's events in Tunisia".
- Worst attack since 2002 -
The attack was the worst on foreigners in Tunisia since an Al-Qaeda suicide bombing of a synagogue killed 21 people on the island of Djerba in 2002.
US Secretary of State John Kerry denounced the "wanton violence".
British Prime Minister David Cameron said he was "appalled" by the attack and French President Francois Hollande expressed "solidarity".
Palestinian Islamist movement Hamas described the shooting as "a crime against humanity".
A Japanese survivor described how she and her mother were shot in the hail of bullets.
"I was crouching down with my arms over my head, but I was shot in the ear, hand and neck," 35-year-old Noriko Yuki said from her hospital bed in comments aired by Japanese broadcaster NHK.
"My mother beside me was shot in the neck."
Museum employee Dhouha Belhaj Alaya said she heard "intense gunfire" around noon.
"My co-workers were screaming 'Run! Run! Shots are being fired!'" she told AFP. "We escaped out the back door with co-workers and some tourists."
Tunisia has taken pride in forming a democratic government and achieving stability since the Arab Spring -- in marked contrast to countries such as Egypt and Libya.
But dozens of police and military personnel have been killed or wounded in attacks blamed on Islamist militants.