National counter-terrorism police chief Mark Rowley named two of the three slain assailants as Khuram Butt and Rachid Redouane, revealing that Butt had been known to security services.
The aftermath of Saturday night's rampage, which left seven dead and dozens wounded, dominated the campaign trail ahead of Thursday's general election.
Labour leader Jeremy Corbyn said he would support calls for May to quit, as she had overseen a sharp reduction in police numbers in her past job as interior minister.
The attack, claimed by the Islamic State group, saw three men wearing fake suicide vests use a white van to mow down people on London Bridge and then slash and stab revellers enjoying a Saturday night in the bustling Borough Market area. Armed police reacted swiftly, killing the attackers within eight minutes with 50 shots. Butt was 27 and a British citizen born in Pakistan.
London attack: What we know
The second attacker was named as 30-year-old Rachid Redouane, who police said claimed to be Moroccan and Libyan, and also went by the name Rachid Elkhdar with a different date of birth.
Both men lived in the same area of east London. Police said they were still working to establish the identity of the third attacker.
Late on Saturday the three attackers drove south across London Bridge, mowing down pedestrians before stabbing bystanders in the nearby Borough Market area.
Khuram Butt: An extremist with a friendly family image
Khuram Shazad Butt, named Monday as one of the three London attackers, was known to Britain's MI5 intelligence service and even featured in a documentary entitled "The Jihadis Next Door".
Butt had sparked alarm for his extremist views and allegedly had links to a jailed radical preacher, yet neighbours at what was reportedly his home in Barking, east London, remembered a friendly family man.
A 27-year-old British citizen born in Pakistan, Butt was shot dead by police along with his two accomplices after killing seven people and leaving dozens wounded on Saturday night.
Butt had worked for the London Underground for six months last year as a trainee customer services assistant, Transport for London told the BBC. British media said he had also worked at the fast food chain KFC and was a keen football fan.
But he appeared in a Channel 4 documentary entitled "The Jihadis Next Door" about British extremists that was broadcast last year, according to local media. And his extremist views had reportedly prompted numerous people to go to the authorities.
They include a teenager who went to the same gym as Butt who said he was approached to join the Islamic State (IS) group, and a woman who told police she feared he was trying to radicalise children, The Times said.
Mohammed Shafiq, head of the Ramadhan Foundation think tank, said he was verbally abused by Butt in 2013 outside the British parliament. He said Butt had called him a traitor in Arabic and accused him of being a government stooge.
Shafiq said Butt was at the time with radical preacher Anjem Choudary, the former head of a now-banned group called al-Muhajiroun. Choudary was jailed for five-and-a-half years in September for encouraging support for IS militants.
Among those radicalised by Muhajiroun were the suicide bombers who killed 52 people on London's public transport system in July 2005, and the men who murdered soldier Lee Rigby in the capital in 2013.
UK PM May says to deploy armed forces after attack as threat level raised
"I am not surprised that Khuram Butt carried out the terrorist attack and there are serious questions for the authorities," Shafiq said. But neighbours in east London described the suspected assailant as an affable father-of-two.
"We saw him many, many times around here," said Salahudeen, a 40-year-old driving instructor.
"He used to be friendly but suddenly... he wasn't acting as normal. He wasn't aggressive; he used to chat but lately he was just 'hi' and 'bye'," he told AFP.
Salahudeen said the man had two children -- a baby girl and a boy who used to be taken to the park by his father to play football.
Michael Mimbo, 25, said the suspected attacker was nicknamed "Abz" at the local gym.
Police on Monday also named 30-year-old Rachid Reodouane as one of Butt's accomplices but did not give any details about the third attacker. Redouane claimed to be Moroccan and Libyan and also used the name Rachid Elkhdar, police said.
The attacker was carrying an Irish residency card and had lived in Dublin, Ireland's national broadcaster RTE reported Monday, citing security sources. RTE said he had married a British woman in Ireland in 2011 before moving to Britain. The police suspect his wife is currently in custody in Britain, it said. The Guardian reported that he was a pastry chef.
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