Indeed, McAvennie has spoken warmly of the 31-year-old, who he thinks still has the ability to do a job in the Premier League.
In the Scottish Premiership, though, the former West Ham and Celtic man feels Cole can be as big of a hit, singling out his unselfish nature as a player as his best attribute.
McAvennie is quoted as saying by the Express: "I think he would be a good signing if Celtic decide to do the deal. You don’t play in the Premier League for as long as he has if you are not a top player.
"He will score goals. He scored goals in the English Premier League and I have no doubt he would score even more goals in the Scottish Premiership.
"But, for him, it is more about the team than any individual glory. He is a team player and one the fans would warm to. He could still do a job in the English Premier League, so that tells you the sort of quality he has."
It was always a little unthinkable that Cole could return to West Ham for a third spell, of course, even though the east London side were in pursuit of a new striker this summer.
In the end, they went for Nikica Jelavic but Cole's surprise arrival at Parkhead over the weekend could see the 31-year-old make a very good career move of his own.
Celtic should be a good destination for Cole and, according to McAvennie at least, the striker will be a good choice for the Bhoys. Of the other free agent options available to them, only Georgios Samaras stands out...