"Supo has become aware of more serious terrorism-related projects and plans in Finland", it said.
"The suspect's identity and nationality are yet to be determined", he added. The attacker is being treated in hospital, with police being unable to question him yet.
He has been taken into custody after being shot in the leg, police say.
Describing the incident, an eyewitness told BBC, "I saw police shoot a person, a man I think".
"Police have told us not to go to the city center, so we are in this coffee shop a few blocks away", said Vanessa Deggins, an American studying business at one of Turku's three universities, told The Associated Press.
Finnish broadcaster YLE said several people were seen lying on the ground in Puutori square in central Turku.
It said it had no reports of any Irish citizens being affected by the attack. This could change, however, should new evidence come to light.
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That it was a terror attack was not yet confirmed by authorities, however, as news broke on August 18.
A video posted online appears to show the moments after the incident. Shouting is heard in the background. However, it was later reported that the man was likely a lone attacker.
The probe is now focusing on determining the motive of the attacker, with police anxious to question him once his medical condition improves.
There were reports of multiple people armed with knives and gunshots being fired in the city centre earlier this afternoon.
Police have revealed that both of the dead were Finnish.
The Turun Sanomat newspaper reported that at least one person had been killed.
"The act had been investigated as murder, but during the night we received additional information which indicates that the criminal offenses are now terrorist killings", police said in a statement. At least one remains at large.
Finnish media, via this CNN clip, is reporting a major stabbing incident in Turku. There was some talk about a knife attack.