The 53-year-old victim, who had reportedly travelled to the UK from Ireland for the match, sustained a serious head injury in the attack.
He is being treated at a specialist neurological centre, where he is said to be in critical condition.
Two Italian nationals, both men and aged 25 and 26, have been arrested for attempted murder, Merseyside Police said.
Witnesses say he was injured when a fight broke out between Roma and Liverpool fans outside a pub shortly before kick-off. He was reportedly hit with a belt and fell to the ground.
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Liverpool FC, who won the Champions League semi-final first-leg match 5-2, said they were "shocked and appalled" by the incident.
Roma also condemned the violence.
"AS Roma condemns in the strongest possible terms the abhorrent behaviour of a small minority of travelling fans who brought shame on the club and the vast majority of Roma's well-behaved supporters at Anfield after getting involved in clashes with Liverpool supporters before last night's fixture," the club said.
European football's governing body UEFA said it was "deeply shocked by the vile attack", adding it was waiting to receive full reports before deciding on potential disciplinary charges.
A total of nine men were arrested for various offences before and after the match, including assault, possession of offensive weapons, criminal damage and being drunk and disorderly.
Photo: Oscar Federico Bodini/Flickr