The 18-year-old, who has scored four goals this season, is understood to have attracted the interest of a host of Premier League clubs. However, Davis has moved to play down suggestions of an offer from the Gunners.
"I've had no enquiries about Max, only people watching, along with a lot of our other players, so it's only a rumour," he told talkSPORT.
"He's not ready to go anywhere. He's just finding his feet with us at the moment. He's got a bit to learn yet before he leaves this football club.
"When the situation is right he will leave to go in somebody's first-team. There is no real point him going and playing into a development group when he's playing regularly here. This is the best education he can get."
Crewe currently sit 19th in the League One table.