The 40-year-old was injured in the attack in Paisley, Renfrewshire, in an incident being treated as attempted murder by police.
It is understood officers are investigating the possibility the incident was football-related.
Police Scotland said on Tuesday they are following a positive line of inquiry.
A spokeswoman for the force said: “At around 2.55pm on Sunday, emergency services responded to reports of a 40-year-old man having been seriously assaulted in Westburn Avenue, Paisley.
“The injured man was taken to the Royal Alexandra Hospital in Paisley and then transferred to the Queen Elizabeth University Hospital in Glasgow for treatment.
“Hospital staff describe his condition as stable.”
Following the Scottish Cup semi-final tie between the two Glasgow clubs, police revealed nine men aged between 18 and 56 were arrested at the game and will be reported to the procurator fiscal for “minor public order offences” such as breach of the peace, sectarian singing and pitch incursion.
Rangers won the match on penalties after extra time finished 2-2 at Hampden Park.