The 40-year-old man arrested for "attempted murder" in the early hours of Thursday has connections "with a group of ultras" linked to Atletico, according to police in the Spanish capital.
The attack happened on Wednesday evening close to Atletico's Wanda Metropolitano stadium, shortly before the team's 2-1 defeat to Sevilla in the Copa del Rey.
The two men seem to have become involved in a dispute in a bar not far from the ground in the east of the capital.
The victim was admitted to hospital in a serious condition after being stabbed three times.
Aged 22, he suffered injuries to his back, his shoulder and his thigh. He has undergone surgery and his life is not in danger.
Police did not name the suspect but Spanish media reports said the man had previously been implicated in the death of a Real Sociedad fan before a game against Atletico in 1998.
That case was later dropped due to a lack of evidence.