Cole started his career at Chelsea and had loan spells at Wolves, Charlton and Aston Villa, but he has been without a club since the Hammers freed him in May.
McAvennie said: “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.”
Cole, who won the last of his seven caps in 2010, also has options in Major League Soccer and the English Championship, but he would love to test himself in the Europa League with Celtic and get the opportunity to win domestic trophies.
Cole said: “Glasgow has surprised me, I’ve never been to Scotland before so, hopefully, we can do a deal.
“I was almost going to the MLS instead of coming here, so, hopefully, I do like it here.”
