All three attacks have been claimed by the Islamic State group.
In at least two of the cases, the men were said to be known to police, raising pointed political questions just two days before a national election.
Zaghba was "not a police or MI5 subject of interest", the police said in a statement, but according to Italian newspaper Corriere della Sera, Italian intelligence had tipped off their British counterparts when he tried to go to Syria previous year via the city of Bologna.
"I don't know, I don't know", said Jibril Palomba, whose wife had reported Butt.
Police stopped Zaghba because he was only carrying a one-way ticket and a backpack, and they suspected he meant to reach Syria via Turkey to join ISIS.
Italy's anti-terrorist force DIGOS believed he was trying to join Islamic State militants there.
Then, his passport and cellphone were sequestered after he tried to board a flight for Turkey with a one-way ticket and a small backpack.
Relatives of the Spanish victim in the London Bridge attack last week say they would like to take the body of Ignacio Echeverria back to Spain as soon as the autopsy scheduled for Friday is conducted.
Zaghba was the son of an Italian mother and a Moroccan father who had separated. His mother is Italian and still lives there.
She told reporters on Wednesday that he had changed in the past year.
After police named two of the attackers and revealed that one was previously known to security agencies, May's Conservative Party faced further questions about her record overseeing cuts to police numbers.
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He received an European Union residency permit from Ireland and the couple lived there until at least the end of 2015.
Up to two-thirds of all terror plots involving United Kingdom nationals at home and overseas in the previous two decades had been linked to the group according to the sources, including the 2005 London bombings, the murder of British soldier Lee Rigby in east London in 2013 and a copycat plot in August 2014 to behead another British soldier in the capital.
The Guardian reported that he worked as a pastry chef.
Butt was shown unfurling a black ISIS flag on a show called The Jihadis Next Door, screened four months before he got a customer services job at Transport for London in May 2016. Police typically shift investigations to a lower priority level in such circumstances to free up resources for what they regard as more imminent threats.
But his extremist views had reportedly prompted numerous people to go to the authorities.
Zaghba, Pakistan-born British citizen Butt and Redouane, 30, who claimed to be Moroccan-Libyan, launched a rampage around London Bridge and Borough Market on Saturday night. "I don't know him any more".
"We always kept an eye on who he was friends with and checked that he was not mixed up with the wrong people", she told the magazine in her only interview since her son was identified.
Witnesses last saw 39-year-old Echeverria last Saturday lying on the floor after confronting the assailants near London's Borough Market with his skateboard.
British media said he had also worked at the fast food chain KFC and was an avid fan of the Arsenal football team.
He said his extremist nephew had only visited Pakistan twice, but that he had noticed a change in the young man.
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