A Libyan man has been arrested on suspicion of launching the rampage as groups of friends enjoyed the evening sunshine on Saturday.
Security officials told The Independent that there was a strong possibility that the motivation was terror-related, and investigators are also examining whether the suspect had any mental health issues.
Thames Valley Police said the attack had not officially been declared a terrorist incident and was currently being treated as a murder inquiry.
Witnesses heard the knifeman shouting slogans as he carried out the attack at around 7pm.
Police said a 25-year-old man from Reading was detained at the scene and was arrested on suspicion of murder. He remains in custody.
Detective Chief Superintendent Ian Hunter, said: “We have made one arrest and at this stage we are not looking for any other people in connection with this incident. There is no intelligence to suggest that there is any further danger to the public, however we urge people to remain vigilant and report anything suspicious by calling police.
“This is not currently being treated as a terrorism incident, however officers are keeping an open mind as to the motivation for the incident and are being supported by colleagues from Counter Terrorism Policing South East.”
The prime minister and home secretary were informed as soon as possible after the killings, in line with major terror incidents.
Boris Johnson tweeted: “My thoughts are with all of those affected by the appalling incident in Reading and my thanks to the emergency services on the scene.”
Priti Patel called the attack “senseless”, adding: “My heart, prayers and thoughts are with all of those affected and to the people of Reading who will be deeply shocked and concerned by this terrible incident.
“I am in contact with the police and will ensure they have all the support they need.“
If it is confirmed to be a terror attack, it would be the deadliest to hit the UK since the June 2017 London Bridge attack.
The attack happened in Forbury Gardens, which had hosted an unrelated Black Lives Matter demonstration earlier in the day and was filled with people enjoying warm weather.
A personal trainer who fled the attack said the knifeman had shouted “unintelligible words” before stabbing victims in the neck.
“The park was pretty full, a lot of people sat around drinking with friends, when one lone person walked through, suddenly shouted some unintelligible words and went around a large group of around 10, trying to stab them,” said Lawrence Wort, 20.
“He stabbed three of them, severely in the neck, and under the arms, and then turned and started running towards me, and we turned and started running.
“When he realised that he couldn’t catch us, he tried to stab another group sat down, he got one person in the back of the neck and then when he realised everyone was starting to run, he ran out the park.”
Graphic footage from the scene showed the three victims lying injured metres apart on the grass, surrounded by members of the public as police carried out emergency first aid.
Officers could be heard calling for a defibrillator, as CPR was conducted on one victim.
Air ambulances and several other ambulances were called, as well as teams from London and a hazardous area response team.
The suspect was chased and pinned to the floor by police officers. He was searched for weapons before being taken into custody.
Heavily-armed Counter Terrorist Specialist Firearms Officers later searched a flat in Basingstoke Road, in the Reading suburb of Whitley.
Residents were evacuated from the block, which sits just under two miles from the scene of the stabbing.
Families with children were left gathered at the roadside as the operation continued into the early hours of Sunday morning, with a helicopter hovering overhead.
A local woman, who did not want to be named, told The Independent the flat where the suspect lived was council-owned and used as temporary accommodation, including for people recently released from prison.
“First there were armed response cars, and then unmarked vans with armed police wearing balaclavas turned up,” she added.
At the beginning of the UK’s coronavirus lockdown, the closure of large venues and transport hubs caused counter-terror police to assess that the risk of mass stabbings had reduced, because of a reduction in crowded places.
But at the same time, there were concerns that the effects of online radicalisation could be more pronounced as people became more isolated and spent more time on the internet.
The head of the Prevent counter-extremism programme previously told The Independent referrals had fallen sharply during the pandemic, raising fears that potential threats were not being spotted by teachers, probation workers, NHS staff and other agencies forced to reduce contact with people.
Isis has incited its supporters to carry out low-technology terror attacks using knives and vehicles around the world.
The group released advice on picking “soft targets” containing crowds of people but low security.
The tactics have been used in several recent terror attacks in the UK, including at Fishmongers’ Hall near London Bridge in November and Streatham, southwest London, in January.
As of 1am on Sunday, no claim of responsibility had been issued from Isis’ official propaganda channels.
Callers to Thames Valley Police’s 101 non-emergency service were greeted by an automated message saying officers were dealing with a “major incident which has absolute priority for all our resources” as the investigation continued into Sunday.
Under coronavirus laws, gatherings of up to six people in gardens and public spaces are now legal in England and masses of people have flooded into parks.
Nieema Hassan, an organiser of the anti-racism Reading protest, posted on Facebook that demonstrators had dispersed by the time of the attack.
She added: “In terms of the protest and people that attended from Black Lives Matter, we’re all safe, none of us are affected.
”I have goosebumps knowing that this has happened only a few hours after we all peacefully came together today to talk about change and equality and it’s pretty heartbreaking that this day has been tarnished with violence.“
Police confirmed that the incident was not believed to be related to the earlier Black Lives Matter protest.