The third London Bridge terrorist has been named as Youssef Zaghba, an Italian national of Moroccan descent.
Police said they had to prioritize resources on suspects who were believed to be preparing an attack or providing active support for one.
17 people have now been arrested in connection with Saturday's attack and five people remain in custody.
An Italian prosecutor says Zaghba told authorities after being stopped past year at Bologna's airport that he "wanted to be a terrorist", but then quickly corrected himself.
He said Zaghba was always tracked by Italian intelligence officers while in the country and that United Kingdom authorities were informed.
The Italian reports said Zaghba was intercepted at Bologna airport a year ago as he was about to board a plane for Turkey, with the apparent intention of joining Islamic State militants fighting in Syria. Zaghba's Italian citizenship prevented such an expulsion, Italian daily Repubblica reported.
Zaghba was reportedly working in a London restaurant and had not been seen in Italy since 2016.
Scotland Yard has said Zaghba was not a police or MI5 "subject of interest". A new search was underway Tuesday in Ilford, just north of Barking, as authorities tried to determine whether the group had accomplices.
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.
Detectives searching for a French man who disappeared during the London Bridge terror attack have recovered a body from the River Thames.
Security has become a key issue in the run-up to Thursday's general election.
Britons cast ballots in election marred by terror attacks
Corbyn, meanwhile, accused Conservatives of undermining Britain's security by cutting the number of police on the streets. It rained intermittently across much of the country on polling day, but experts said it would likely not affect turnout.
Butt would help run high-octane mixed martial arts classes, and would allegedly use the gym to approach young pupils for radicalization.
London attack ringleader Khuram Butt was identified as a significant potential threat in an investigation that started in 2015, British counterterrorism sources told CNN on Tuesday.
Police in Ireland arrested one man in Limerick on Monday night and a second in Wexford 24 hours later.
The marksmen are charged as they train their guns on the attackers, with one knifeman, who appears to be Butt, swinging his weapon wildly towards an officer. "However, there was no intelligence to suggest that this attack was being planned and the investigation had been prioritized accordingly".
As interior minister from 2010 to 2016, May oversaw a drop of 20,000 in the number of police officers in England and Wales, which Labour leader Jeremy Corbyn has said should never have happened and warranted her resignation.
The number of police officers in England and Wales fell by nearly 20,000 between 2010 and 2016 - years when May, as home secretary, was in charge of policing. Borough Market, a popular gathering place, remains closed as more evidence is gathered.
A minute's silence was observed in Britain at 11 a.m. local time (1000 GMT) in memory of those killed during the attack.
Meanwhile, the family of Spanish banker Ignacio Echeverria have confirmed he was killed in the attack after trying to fend off a terrorist with his skateboard.
It was lowered once intelligence agencies were comfortable this wasn't the case.
His mother, Valeria Collina, a convert to Islam who is separated from the attacker's Morocco-based father, told Italian weekly L'Espresso that she thought her son had been radicalised by a combination of online propaganda and contacts in London.
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