The former City forward thinks the manager will be under pressure come the end of the season.
City were beaten 4-2 by fierce rivals Manchester United on Sunday afternoon in the Manchester derby, leaving them fourth in the table and with little chance of defending their title.
It has been a disappointing season in total from last year’s champions, with them once more failing to defend their title while they have also exited the Champions League and cup competitions as well.
That has led to increased pressure on manager Pellegrini, and Quinn says he is definitely in the firing line.
“His head’s on the block and he’s in the firing line,” said Quinn.
“How they finish off is important. They’ve got to show they really want to play for this guy and in turn he’s got to make sure they’re up for and they want to stay.”
The City team's poor performances this season have led to pressure for not only manager Pellegrini but the rest of his squad also, with some pundits suggesting there needs to be a major overhaul of the team.
Some of City’s more consistent performers over the years have failed to impress this season, such as Yaya Toure, while signings made in the summer look unlikely to be long-term players at the club.
Now, Quinn has suggested there needs to be changes in the squad during the summer.
“I’m not so sure that that great squad they had last year and that great passion is in the dressing room now,” he added.
“It’s not a particularly comfortable place for Pellegrini all round, but if there are people in that dressing room who aren’t pulling their weight, he’s got to move them on. “