The boy on Tuesday brought a relative's revolver to the Viertola school near Helsinki and shot the three other 12-year-olds and threatened several others.
He had transferred to the school at the beginning of 2024, investigators said.
Flags flew at half-mast across the country on Wednesday as it mourned the boy's death.
The two other pupils remained in hospital.
"We just found out today that there was this bullying behind the tragedy," lead investigator Detective Chief Inspector Marko Sarkka told Reuters.
The preliminary police probe had confirmed that the boy was a target of bullying, investigators said in a separate statement.
Sarkka declined to elaborate.
Police have not said whether the attacker sought to target any specific individuals.
The permit for the revolver used in Wednesday's attack belonged to a relative of the suspect, according to police.
It was not immediately clear how the shooter had got hold of the weapon.
"This matter is being investigated by the police as a separate firearms offence," investigators said in a statement.
