One of the London Bridge attackers was able to enter the UK, despite being placed on an EU-wide watch list.
Youssef Zaghba, a 22-year-old Moroccan-Italian man who lived in east London, was named as the third attacker.
He was stopped at an Italian airport on his way to Syria last year and was put on an EU-wide database but was not prosecuted, reports say.
Zaghba, Khuram Butt and Rachid Redouane killed seven people and injured 48 others during the attack on Saturday.
So far four of those killed in the attack have been named: Australian Kirsty Boden, Canadian national Chrissy Archibald, James McMullan, from Hackney, and French national Alexandre Pigeard.
French Foreign Minister Jean-Yves Le Drian confirmed on Wednesday that a second French citizen had died.
The three attackers drove a hired van into pedestrians on London Bridge at 21:58 BST before stabbing people in the area around Borough Market.
Armed officers killed all three within eight minutes of receiving a 999 call.
Speaking at her house in Bologna, Italy, Zaghba's mother told the BBC that she believed her son was radicalised in the UK.
She claimed her son was under surveillance when he was in Italy, but she questioned why this was not the case in the UK.
An Italian police source has confirmed to the BBC that Zaghba had been placed on a watch list, which is shared with many countries, including the UK.
In March 2016, Italian officers stopped Zaghba at Bologna airport and found IS-related materials on his mobile phone.
He was then stopped from continuing his journey to Istanbul.
The BBC understands he was not prosecuted but was listed on the Schengen Information System, an EU-wide database which includes details of potential suspects.
When Zaghba entered Britain, staff at passport control should automatically have been alerted by the Schengen system, BBC home affairs correspondent Danny Shaw said.
"One unconfirmed report suggests that did happen, apparently when Zaghba arrived at Stansted Airport in January - but that border staff still let him in," he said.
When asked if this was the case, Work and Pensions Secretary Damien Green, a former Home Office minister, told BBC Radio 4's Today programme that if someone's passport comes up on the Schengen system the person should be stopped at the border.
"I obviously don't know what happened in this case," he said.
"It would be wrong to comment on an individual case while there is a very serious continuing police investigation going on."
On Wednesday, the Metropolitan Police said a 30-year-old man was arrested on suspicion of terror offences in Ilford, east London.
On Tuesday a 27-year-old man was arrested in Barking on Tuesday in connection with the investigation.
Scotland Yard has been criticised for the way it handled intelligence about Butt, who had been investigated by police and MI5 and featured in a Channel four documentary on extremism.
Police said Pakistan-born Butt, 27, from Barking, had been subject to an investigation in 2015, but there had been no suggestion an attack was being planned.
Redouane, 30, was a chef who also used the name Rachid Elkhdar and police said he claimed to be Moroccan-Libyan. He married a British woman in Dublin in 2012 and lived in the city's Rathmines area.
A man was arrested in Limerick, in the Irish Republic on Tuesday over the discovery of ID documents in Redouane's name. He was later released without charge.
A second man, who is in his 30s, was arrested on Tuesday evening, with the Garda saying it was also related to Redouane. He is being held in Wexford.
Entering the final day of election campaigning, Theresa May said she will change human rights laws if they "get in the way" of tackling suspected terrorists.
The PM said she would make it easier to deport foreign terror suspects and "restrict the freedom and movements" of those that present a threat.
Sir Keir Starmer, the shadow Brexit secretary and director of public prosecutions between 2008 and 2013, said existing human rights laws did not stand in the way of taking action against suspected terrorists.
"If we start throwing away our adherence to human rights... we are throwing away the very values at the heart of our democracy," he told BBC Radio 4's Today programme.
Former SAS commander Col Tim Collins has said intelligence "isn't good enough", adding that police needed to recruit informants from within the Muslim community and appoint Muslim officers.
"MI5 and the police have to recruit sources, informants," he told Today. "Intelligence from within the community is improving, but there's a lot more to be gleaned."
Who were the victims?
Four of those killed in the attack have been named.
Australian Kirsty Boden, 28, worked as a senior staff nurse at Guy's and St Thomas' Hospital in London. Her family have described her in a statement as an "outgoing, kind and generous person".
"We are so proud of Kirsty's brave actions which demonstrate how selfless, caring and heroic she was, not only on that night, but throughout all of her life," they added.
The hospital said that Ms Boden was "an outstanding nurse and a hugely valued member of the staff team in Theatres Recovery, described by her colleagues as 'one in a million' who always went the extra mile for the patients in her care".
Canadian national Chrissy Archibald, 30, was the first victim to be named. Her family said she had died in her fiancé's arms after being struck by the attackers' speeding van.
The family of 32-year-old James McMullan, from Hackney, east London, say they believe he also died.
Mr McMullan's sister said he was believed to be among those who died, after his bank card was found on a body at the scene.
Two French nationals were also killed in the attack, according to foreign minister Jean-Yves Le Drian.
The BBC understands one of them to be Alexandre Pigeard, 27.
Manager of the Boro Bistro, Vincent Le Berre, told the Brittany news outlet Le Telegramme how his colleague was attacked in a bar near Borough Market.
"I managed to escape him, but my friend Alexandre did not have that chance," he said. "He was hit in the neck with a knife."
Mr Le Drian confirmed on Wednesday that a second French citizen had been "identified among those who have died".
The identity of the victim has not been released.
The Met have set up a casualty bureau on 0800 096 1233 and 020 7158 0197 for people concerned about friends or relatives.