Allardyce has taken over the reigns at the Stadium of Light following Sunderland's dismal start to the Premier League season, which sees them without a win and at the bottom of the ladder.
Cattermole says the 61-year-old's track record in similar situations has filled him with confidence.
"I always liked Sam's squads, when you played his clubs you always felt you were playing a team of men," Cattermole told The Guardian
"I always thought I'd like to be in that team.
"Sam fills you with confidence, he's clearly a very good manager. Whether he gets time, whether he gets the players he wants, that's out of our hands.
"But we need stability and that comes from a manager being able to implement his philosophy.
"That can only happen if we make sure he is still in a job come July. We need to keep hold of him and let him build the club.
"We have to trust Sam Allardyce, give him the reins and say: 'Go and do your stuff'. At the moment, though, this isn't Sam's squad so we must do the best we can for him."