Crawley director of football Steve Coppell is understood to have spoken to Holdsworth and Shots chairman Kris Machala confirmed he turned down an "overwhelming" offer.
"Dean is highly regarded within the game and could have taken the opportunity to move. He received an overwhelming offer that would have been beneficial to him both on a personal and professional basis," Machala told the club's official website.
"I am delighted Dean has chosen to remain as manager of Aldershot Town Football Club and I have to place on record my sincere appreciation to an individual who has proved his loyalty and integrity.
"We have a close bond throughout the club and are all genuinely excited as we enter the 20th year of Aldershot Town FC."
Holdsworth himself added: "I have got a lot of ambition but at this moment in time that is making Aldershot Town Football Club a better place to be and work - it is important that people see my loyalty."