Police named the man as Khalid Masood on Thursday, saying he had a string of criminal convictions.
Masood, 52, was born in Kent to the southeast of London and had been most recently living in central England, London police said.
"Masood was not the subject of any current investigations and there was no prior intelligence about his intent to mount a terrorist attack," a police statement said.
"However, he was known to police and has a range of previous convictions for assaults, including GBH [grievous bodily harm], possession of offensive weapons and public order offences."
He had not been convicted previously for any terrorism offences, it said.
In a statement to the House of Commons on Thursday, May said Masood was once investigated by intelligence officers over concerns of "violent extremism".
"He was a peripheral figure," she said. "The case is historic, he was not part of the current intelligence picture."
The Islamic State of Iraq and the Levant (ISIL) group claimed responsibility on Thursday for the attack. It said on its Amaq website the attacker "carried out the operation in response to calls for targeting citizens of the coalition" of countries fighting ISIL in Syria and Iraq.
It was not possible for Al Jazeera to independently confirm the claim, which did not offer any details of the attack or name Masood, casting doubt on whether there is any direct link between ISIL and the London killings.
Joseph Downing, from the London School of Economics, expressed scepticism over ISIL's claim.
"To me this is something quite common over the last couple of years, over the terrorist attacks in Europe, that ISIL jumps on the bandwagon in the most horrific way and says 'yeah, this our soldier', when there's actually no link between the person carrying out the attack and any particular group," he told Al Jazeera.
Some 40 people were wounded in the attack, 29 of whom were being treated in hospital, according to police. Seven were still in critical condition.
May said those wounded in the attack included 12 Britons, three French children, two Romanians, four South Koreans, two Greeks, and one each from Germany, Poland, Ireland, China, Italy and the United States.
Three police officers were also wounded.
The victims included Keith Palmer, a 48-year-old police officer who was stabbed to death, and two members of the public - a woman in her mid-40s and a man in his mid-50s.
The fourth death was the attacker.
Earlier on Thursday, police said eight people had been arrested after raids on six homes in London, Birmingham and other parts of the country in their investigation into the attack, in which a man ploughed into pedestrians in a car and then went on a stabbing spree before being shot dead.
Mark Rowley, acting deputy commissioner at the Metropolitan police, also revised down the number of victims to three from four.
"It is still our belief that the attacker acted alone [and] was inspired by international terrorism," Rowley said.
Al Jazeera's Barnaby Phillips, reporting from London, said: "The absolute priority of the police at this point in time would be to know what sort of accomplices, if any, the assailant had. What sort of assistance, if any, did the assailant have and whether he belonged to any sort of network."
Rowley said he had no specific information about any further risk to the public, but repeated that more officers were on the streets - armed and unarmed - and that many had leave cancelled or were working extended hours.
Some of the wounds suffered by the victims were described as "catastrophic". One woman was pulled out alive from the River Thames with serious injuries by port authorities.
"We saw a black vehicle at full speed and it ran down a number of people. I could see people flying all around," tourist Babi Nagy told Al Jazeera. "Immediately it came to mind this was a terrorist attack."
Polish politician and journalist Radoslaw Sikorski posted a video on Twitter of the aftermath on the bridge, showing several wounded people lying on the ground.
Another witness said he saw victims scattered along the street.
"As I was walking up the steps, there was a man who had fallen and medics were taking care of him. There was a lady who was also stabbed or shot. There was a lot of blood," Martin Pearce told Al Jazeera at the scene.
The last major attack to hit London was in July 2005, when a coordinated series of bomb blasts targeted its public transportation system during rush hour. The bombings killed 52 people and wounded more than 700 others.
Source: Al Jazeera and news agencies