Sky Italia and the BBC reported that the attack happened around 2.30am local time (1.30am GMT) in the Prati district of the city, around two milers away from the Stadio Olimpico, the scene of Celtic's triumph.
BBC reported that a man of 20 was taken to a local hospital in the city for a stab wound in his lower back, but he was later released.
Before the game, two fans were stabbed by masked 'ultras' outside an Irish bar in Rome.
They suffered leg-wounds but their injuries were not believed to be life-threatening.
One of the men remains in hospital, and police are hunting for the perpetrators of all three attacks.
Separately, Celtic fans were attacked after their shuttle bus on the way back from the stadium broke down.
Tensions between the two sets of fans after the granddaughter of fascist dictator Benito Mussolini demanded Celtic fans be prosecuted for a banner depicting her grandfather's hanging body.