Indeed, the former Celtic man feels the 24-year-old edges out his Sunderland counterpart due to his superior movement.
And McAvennie has urged Scotland boss Gordon Strachan to turn to Griffiths when his side return to action, as Scottish fans will be urging him to give the Bhoys forward a nod.
McAvennie is quoted as saying by the Mail: "It will be a great headache for Gordon Strachan by the time Scotland are back playing next year. If he wants to play Fletcher, the whole country will want Griffiths.
"I know Fletcher has been scoring but I genuinely think Leigh would be a better shout because of his movement.
"I just don’t buy into the talk that he isn’t ready to step up to international level. He should be given a chance when it comes around again. Let him prove to everyone he can be the man. He is an absolute pest and I like that in a striker."
Fletcher looks as though he has improved of late both for Scotland and Sunderland, while the 28-year-old naturally has experience on his side.
Griffiths, however, is the form striker of Scottish football and, as McAvennie states, has also proven himself in Europe.
The Celtic forward will have to prove himself in Europe again, of course, when Celtic host Molde during their crunch Europa League clash on Thursday night.
Sunderland, meanwhile, will look to Fletcher for another goal or two as the Black Cats host in-form Southampton at the Stadium of Light on Saturday afternoon.