The official declined to reveal the identity of the detained "cell leader," citing security reasons.
"The cell met months ago with wanted Algerian terrorist Luqman Abu Sakhr, the leader of Uqba ibn Nafi militant group, which recently swore allegiance to ISIL," Chelli told AA.
At least 23 people, including 20 foreign tourists, were killed last Wednesday when gunmen stormed Tunis' Bardo Museum.
Tourists from Italy, Japan, Poland and Spain were among those killed in the attack, responsibility for which was later claimed by a ISIL-affiliated group.
On Monday, Tunisian Interior Minister Najem Gharsalli sacked several senior security officials over the attack.