They said Pakistan-born Khuram Butt, 27, of Barking, London, had been known to police and MI5 but there had not been any intelligence about an attack.
The other attacker was Rachid Redouane, 30, from Barking, who police said had claimed to be Moroccan-Libyan.
The attackers were shot dead by police. All 12 people arrested after the attack have now been released without charge.
The seven women and five men were arrested in Barking on Sunday following the attack in which seven people were killed and 48 injured.
The attackers drove a hired van into pedestrians on London Bridge before stabbing people in the area around Borough Market.
A vigil was held at Potters Field Park near London Bridge on Monday evening to remember the victims.
NHS England said 36 people remained in hospital, with 18 in a critical condition.
Redouane, who was a chef, also used the name Rachid Elkhdar. He had not been known to police.
Metropolitan Police assistant commissioner Mark Rowley said: "Inquiries are ongoing to confirm the identity of their accomplice."
He said the investigation into Butt had begun two years ago but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritised accordingly".
He added: "Work is ongoing to understand more about them, their connections and whether they were assisted or supported by anyone else."
A vigil was held in Potters Field Park, close to London Bridge
Butt featured in a Channel 4 documentary last year about Islamist extremists with links to the jailed preacher Anjem Choudary.
The attacker, who had older siblings and was married with at least one child, could be seen in the programme arguing with police officers in the street.
He went on to work in an administrative role for a company called Auriga Holdings, based in East Ham, which manages Kentucky Fried Chicken outlets.
He had also worked for London Underground for just under six months as a trainee customer services assistant, before leaving in October last year, Transport for London said.
And Butt was the sole director of a now-dissolved company called Kool Kosmetics.
At the vigil
(L to R) Badeiyah Islam, 14, Hanyah Casam, 14 and Zahra Salamut, 15 were among the crowd at Potters Field Park
The vigil began as a sombre and quiet gathering under the trees in Potters Field Park, where people stood in quiet contemplation looking out over the Thames.
So many spoke of their desire to stand together with others and show love.
London Mayor Sadiq Khan's speech roused the quiet crowd of thousands into hearty applause when he spoke of his disgust and defiance over the horror of Saturday's events.
After a minute's silence, many went forward to lay flowers beneath the flagpoles of City Hall.
It was a dignified show of solidarity and strength in the face of fear and hate.
Sources in Dublin said Redouane was understood to have been carrying an identification card issued in the Republic of Ireland when he was shot dead, the Press Association reported.
Irish Prime Minister Enda Kenny said he was not a member of a "small group" being monitored by Irish police in respect of radicalisation.
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 sister of 32-year-old James McMullan, from Hackney, east London, said he was believed to be among those who died, after his bank card was found on a body at the scene.
James McMullan is believed to have been killed in the attack, as well as Chrissy Archibald
A French national was also killed in the attack, according to foreign minister Jean-Yves Le Drian.
Among other developments:
- There will be a minute's silence on Tuesday at 11:00 BST in memory of those who lost their lives and all those affected by the attacks
- Barriers have been installed on Westminster, Lambeth and Waterloo bridges following the attack to stop vehicles from mounting the pavement
- The Independent Police Complaints Commission said 46 shots had been fired at the three attackers by eight police officers - three from the City of London force and five from the Met
- The Met Police has set up a casualty bureau on 0800 096 1233 and 020 7158 0197 for people concerned about friends or relatives
It is the third terror attack in the UK in three months, following the car and knife attack on Westminster Bridge in March, in which five people were killed, and the Manchester bombing on 22 May, in which 22 people were killed.