British police said Zaghba had not been a subject of interest for them or the domestic intelligence agency MI5 before he and two others launched their attack.
Saturday night's attack injured dozens, including four police officers.
Italian newspaper Corriere Della Sera said Zaghba was born in Fez in January 1995 to a Moroccan father and Italian mother.
Zaghba was stopped at the airport in Bologna in 2016 when he was trying to get to Syria via Turkey, city prosecutor Giuseppe Amato told broadcaster Radio24 on Tuesday. After that, he apparently divided his time between Morocco and Italy, where his mother lived outside Bologna.
Italian media reported that officials suspected he was bound for Syria and that Italian authorities had alerted their British counterparts about his movements.
Following Saturday night's terror attack in London, police descended upon two addresses in East London and ultimately arrested 12 individuals in relation to the weekend's atrocity. The search in Ilford, just north of Barking, is seeking to determine whether the group had accomplices.
Another attacker, Rachid Redouane, was not known to police.
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As the scare spread through the crowd gathered in the city's Piazza San Carlo, a rush towards the exit points quickly accelerated. One eyewitness said the panic may have been started by the explosion of a loud firecracker that was mistaken for a bomb.
22-year-old Zaghba moved to east London from his birthplace of Morocco and was from a mixed Moroccan and Italian background. The attackers were shot dead by police. Butt did not fall into that category when they last investigated him.
Further to Geraghty's point about "permanently nonviolent radicals", here's counterterror expert Maajid Nawaz on Fox this morning warning that the same jihadi who appeared on the TV show also attacked a member of his staff a year ago - and that police knew about it. Redouane had not aroused any suspicions. He said it was a dream they had sent to him through the Internet. Meanwhile, The brother of Manchester suicide bomber Salman Abedi has been released by the United Kingdom police.
During his time in Britain, Butt once worked for Transport of London as a customer service clerk but failed his probation after a few months on the job after attendance issues.
"Any successful attack is a failure of intelligence services", said Raffaello Pantucci, counter-terrorism expert at the Royal United Services Institute in London.
Questions are being raised about whether British police missed crucial warning signs that could have prevented the attack.
The news raises questions about the police's judgement and increases pressure on Prime Minister Theresa May, who three days before a national election is facing criticism for overseeing cuts to police during her years as interior minister.
He also pledged to hire more police officers, following a drop in 20,000 officers between 2009 and 2016, or around 14 per cent, according to the independent Institute for Fiscal Studies think-tank.
Much of the area around London Bridge remained cordoned off as commuters struggled to work in the driving rain.