Reports emerged of clashes between Roma and Liverpool supporters shortly before kick-off at Anfield for the semi-final first leg.
Merseyside Police confirmed the force was investigating a "serious assault" of a man outside The Albert pub, which is located close to Anfield on Walton Breck Road.
The force made a social media appeal for information during the match and following the game Merseyside Police confirmed two men, aged 25 and 26, had been arrested in connection with the incident.
The victim, a 53-year-old man believed to be a Liverpool fan, suffered a head injury and is in a critical condition in hospital.
"We believe the victim was in Liverpool with his brother for the Liverpool v Roma semi-final and was assaulted during an altercation between Roma and Liverpool fans near to the Albert pub at about 7.35pm," said detective inspector Paul Speight.
"Witnesses report that the victim was hit with a belt and then fell to the ground. Paramedics treated him at the scene and he was taken by ambulance to Walton Neurological Centre, where he is being treated for a head injury. His condition is currently described as critical and his next of kin have been informed.
"The place where this incident occurred is right by the ground and was heaving with fans making their way to the game. I would appeal to anyone who has any information, or mobile phone footage of the incident to contact us urgently, as information they have could be vital to our investigation."
A Liverpool statement confirmed the club will be in close contact with emergency services over the incident.
"Liverpool Football Club is shocked and appalled after a Liverpool supporter was left in a critical condition having been attacked prior to the Champions League fixture against Roma," a club statement said.
"Our thoughts, in the first instance, are with the victim and his family at this very traumatic time. We will be offering them our full support.
"The club has been liaising with the emergency services since the incident occurred and will continue to do so."
Merseyside Police confirmed nine men were arrested for various offences before and after Liverpool's 5-2 win.