Griffiths joined the Hoops from Wolverhampton Wanderers on transfer deadline day, having scored 31 goals during a loan spell at fellow Scottish Premiership side Hibernian.
Despite this, McAvennie believes that the forward does not possess the ability to fire Celtic to European glory.
"I like Leigh as a player and he is a great addition to the squad but something is not right," McAvennie is quoted as saying in the Daily Record.
"He has not been bought to win the league because the league is won and unless I am wrong he is not the out-and-out striker I would like to see here.
"He is not a poacher. He is a helluva player and will create chances for other players, he will link up and he will score great goals. You can't not like him but I still think Celtic need a poacher."
Griffiths has also had spells with Dundee United and Livingston.