Preliminary evidence suggests that the man arrested near Milan this week in connection with March's deadly attack at the Bardo Museum in Tunis was in Italy when the terrorist strike took place.
Morocco-born Abdel Majid Touil, 22, was apprehended late on Tuesday on the basis of an international arrest warrant issued by Tunisia.
Italian police said he arrived in Italy by boat on February 17 and his movements after that date had to be checked.
The Bardo Museum attack, in which 24 people died including 28-year-old Australian-Colombian dual national Javier Cameloan and his Colombian mother, took place March 18.
However, Fabio Bottero, mayor of Trezzano sul Naviglio, a town where Touil attended adult learning courses, told the Radio 24 broadcaster that school registries show that the suspect was in class March 16 and March 19.
"I passed on this information to relevant authorities, who will investigate," Bottero said.
"I confirm that the boy was in class on March 16 and 19," Flavia Caimi, a teacher, said separately.
In Tunisia, an Interior Ministry spokesman said Touil "participated in the Bardo operation by providing logistical support".
The Tunisian news website Al Choruk, citing security investigations, reported that Touil is believed to have met the two main assailants in a central area of Tunis and to have proceeded with them towards the museum.
Touil was due to appear before a magistrate on Friday for the start of extradition proceedings.
The procedure is expected to face some legal obstacles because in Italy suspects cannot be handed over to countries where they might face the death penalty, which Tunisia has.
Antigone, a prisoner welfare lobbying group, said a pledge by Tunis not to execute the man if he was found guilty would not suffice to free up the extradition.
"The only thing that would count would be the abolishment of capital punishment," it said.
Interior Minister Angelino Alfano told Parliament in Rome that Touil was a "serious suspect" but did not elaborate.
