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.
The latest opinion poll, by Survation for ITV, showed the Conservatives' lead narrowing to just one point from six points in the same poll a week earlier.
Police disclosed late on Monday that 27-year-old Khuram Butt, a British citizen born in Pakistan, was known to police and domestic spy agency MI5 but with resources scarce had not been deemed enough of a threat to warrant close monitoring.
Butt had appeared in a documentary called "The Jihadis Next Door", broadcast last year by Britain's Channel 4, as part of a group of men who unfurled an ISIS flag in a park.
In Britain's third ISIS attack in as many months, Butt and two others rammed a van into pedestrians on London Bridge on Saturday evening before running into the bustling Borough Market area where they slit throats and stabbed people.
All three were wearing fake explosive belts. They were shot dead at the scene by officers within eight minutes of police receiving the first emergency call.
A Canadian, a French national and a Briton were among the dead, while other French people, a Spaniard, Australians and a New Zealander were among the 48 who were injured in what Prime Minister Theresa May called "an attack on the free world".
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 her main opponent, Labour leader Jeremy Corbyn, said should never have happened and warranted her resignation.
May hit back that Corbyn himself was weak on the issue of terrorism. He has been criticised for voting against counter-terrorism legislation and expressing reservations about police responding to attacks with "shoot-to-kill" tactics.
Police arrested 12 people on Sunday in Barking, east London, but said late on Monday all had been released without charge. Police were searching an address in Ilford, also in the east of the city, on Tuesday morning. No one had been detained.
The other attacker who has been named was 30-year-old Rachid Redouane, who also went by the alias Rachid Elkhdar and claimed to be Moroccan or Libyan, police said. He and Butt both lived in Barking.
One of Butt's neighbours, Ikenna Chigbo, told Reuters he had chatted with Butt - known locally as "Abz" - just hours before the attack on Saturday and said he appeared "almost euphoric".
"He was very sociable, seemed like an ordinary family man. He would always bring his kid out into the lobby," said Chigbo.
Another neighbour, Michael Mimbo, told Reuters that Butt supported the north London football team Arsenal. One of the dead attackers has been pictured wearing an Arsenal shirt.
Mimbo said Butt had grown a longer beard and worn traditional Islamic dress more often over the two years he had known him, but showed no sign of radicalisation.
"As an individual he was a cool, calm guy. One of my friends would let Abz babysit his daughter," Mimbo said.
The men's rampage followed a suicide bomb attack which killed 22 adults and children at a concert in Manchester two weeks ago, and an attack in March when five people died after a van was driven into pedestrians on London's Westminster Bridge.
Police said they had to prioritise resources on suspects who were believed to be preparing an attack or providing active support for one. Butt did not fall into that category when they last investigated him.
May did not answer repeated questions from reporters on Monday on the police cuts she oversaw but said counter-terrorism budgets had been protected and police had the powers they needed.
The Conservatives' lead over Labour has narrowed markedly from 20 points or more when May called the election in April to a range between one and 12 points now, although they are still widely expected to win a majority.
The narrowing in the polls started before the Manchester and London attacks and appeared to be largely linked to an unpopular policy announcement regarding care for the elderly. The trend has continued since the two attacks, most polls suggest.
(Additional reporting by Alistair Smout and Kate Holton; editing by Guy Faulconbridge)