Police officers opened fire after the man refused to put down the axe, hitting him in the leg, German news agency dpa reported on Sunday, citing Hamburg police.
German media published images of a person lying in the street surrounded by paramedics and police officers.
The incident occurred in the downtown St Pauli area of the northern port city, which was thronged with fans before Sunday's Group D match between the Netherlands and Poland.
The police spokesman said there was no initial indication that it was related to the Euro 2024 game.
German authorities have put police on high alert during the tournament, which began on Friday and runs through July 14, for fear of possible fan violence and extremist attacks.
On Friday, police fatally shot an Afghan man after he killed a compatriot and later wounded three people watching the televised game between Germany and Scotland in a town in eastern Germany.
Police said on Sunday that the motive for that attack was still unclear.
