Prior to the terror attack, 22-year-old Zaghba was working at a Pakistani restaurant in London.
Pakistan-born Khuram Butt, 27, and Rachid Redouane, 30, both from Barking, were the other two attackers. Butt reportedly quoted the Koran in his final WhatsApp message three days before the attack.
In Britain's third Islamist attack in as many months, three men on Saturday rammed a van into pedestrians on London Bridge before running into the Borough Market area, where they slit throats and stabbed people indiscriminately.
All three perpetrators were confronted and shot dead by armed officers within eight minutes of the first emergency call.
Twelve people who were arrested in Barking on Sunday - believed to include relatives of the dead men - have now been released by police.
The ISIS has claimed responsibility for the latest terror attacks, which rattled London.
Investigators confiscated Zaghba's computer but did not have proof of a crime so they had to return the device and pointed him out to London as a possible suspect, Amato said.
Police sources said he was not on the radar of counter-terrorism investigators, but the paper reported that Italian authorities had alerted British and Moroccan authorities about his frequent movements.
Italian news reports said authorities sequestered Zaghba's cellphone and passport when he was stopped at the airport, but that he successfully got them back after a court determined there wasn't enough evidence to accuse him of any terrorism-related crime. He married a British woman in Dublin in 2012 and lived in Rathmines in the Irish capital.
Butt, Redouane and Zaghba.
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He said the investigation into Butt had begun two years ago but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritised accordingly".
Authorities say the threat is unprecedented, with 500 active investigations involving 3,000 individuals in addition to 20,000 former subjects of interest.
Christine Archibald, a Canadian, was also killed in the attack.
"Work is ongoing to understand more about them, their connections and whether they were assisted or supported by anyone else", Rowley said.
Zaghba's mother, who is a convert to Islam, told police she was anxious about her son when she went to collect him after he was stopped at the airport, an investigative source said.
One of Butt's neighbors, Ikenna Chigbo, told Reuters he had chatted with Butt - known locally as "Abz" - just hours before the attack on Saturday and said he appeared "almost euphoric".
A French national was also killed in the attack, according to foreign minister Jean-Yves Le Drian, which the BBC understands to be Mr Pigeard.
The area around Borough Market is not expected to reopen Tuesday.
"I am going to be a terrorist", he told officials after he was stopped.
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