The 22-year-old defender, who is the son of the Crewe boss, suffered the knock in Saturday's 3-0 defeat to Gillingham.
"It is more positive news involving Harry's injury. He has suffered a grade one, which is the minor of those types of injuries. It is the best result we could have hoped for and it will be weeks rather than months," he told the club's official website.
"It looked serious at the time, because he doesn't go down very often and he came off with physio Nick (Oakley) supporting the knee, so we feared the worse, so this is positive news. He should be out between three to six weeks depending on his recovery."
Davis is expected to return towards the end of October.