Seven people were killed on June 3 and 48 injured when Butt and two other men drove a white van into crowds on London Bridge, and then got out and stabbed people in restaurants and bars in Borough Market.
Armed police responded and within eight minutes had shot dead all three attackers.
The Corriere della Sera newspaper said Italian intelligence had made Zaghba's presence and movements known to the British authorities.
But the focus of attention remains on Butt, who had featured prominently in a 2016 Channel 4 documentary about extremists linked to the jailed preacher Anjem Choudary.
The Mayor of London, Sadiq Khan, said the police and MI5 must reveal what they had on file about Butt and Foreign Secretary Boris Johnson said he had appeared to have "slipped through our net" despite being in plain sight.
Butt, who was married with two children, had worked for London Underground as a trainee customer services assistant.
In the TV program he was seen arguing with police officers after displaying a black Daesh flag in a London park.
Butt, Redouane and Zaghba were all living in Barking, east London, at the time of the London Bridge attack.
One man in Barking had reported concerns about Butt to the anti-terrorism hotline in 2015 and another woman had said he was trying to radicalize her children and she was scared of him.
Metropolitan Police assistant commissioner Mark Rowley said an investigation into Butt began in 2015, but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritized accordingly."
Mr. Khan said: "Not unreasonably, these questions are being asked. I'm sure the police will look into what they knew, what they could have done, what they did do and if anything could have been done differently."
Twelve people who were arrested in Barking on Sunday — believed to include relatives of the dead men — have now been released by police. However, a 27-year-old has been arrested in Barking on Tuesday and is being held under the Terrorism Act.
Butt, who was 27 as well and had been born in Pakistan, is understood to have been the attacker who was seen wearing an Arsenal football shirt at the time of the attack.
In the documentary, The Jihadis Next Door, filmmaker Jamie Roberts befriended Butt and several other Islamist extremists in London. Butt reportedly quoted the Koran in his final WhatsApp message three days before the attack.
His WhatsApp profile showed only verses from the Muslim holy book, which read: "Speak justice, speak kindly, speak politely, speak fairly, speak gently, speak graciously, speak not in vain."
Muslim leaders across Britain have denounced the attack, as well as the May 22 Manchester Arena bombing, as perverted distortions of Islam.
Abduttayyeb Hassanali, who took part in a vigil in London on Monday night with many members of the Dawoodi Bohra Muslim community, said: "I'm a British Muslim and I've lived in London my whole life. I wanted to come and show my support and pay my respects."