O'Donnell became the Spireites' first new arrival this week, signing a one-year deal following his release by neighbours Sheffield Wednesday.
Fans' favourite and reigning player of the year Lee is currently the number one at the b2net Stadium but O'Donnell is determined to make him fight for his place.
"I'm not here to sit on the bench, I'm coming here to push on," the 23-year-old told the club's official website.
"But I'm not expecting to get into the side straight away, I will be working hard every day and hopefully I can push Tommy Lee and get into the team. Working hard is something that comes naturally to me."