The 18-year-old striker is the latest graduate from Crewe's famed youth academy, following most recently in the footsteps of Nick Powell, who swapped Gresty Road for Old Trafford in the summer.
The report suggests Manchester United, Liverpool and Newcastle have all scouted the forward in the last few months, but now Arsenal have joined the pursuit.
Former-Arsenal assistant manager Pat Rice is currently acting as a scout for the Gunners on a part-time basis, and the Northern Irishman is helping Arsene Wenger to make up his mind over Football League stars like Clayton.
The 18-year-old frontman has been compared to former-England international Michael Owen for his blistering pace and sharp eye for goal, and Arsenal have reportedly watched the youngster in two games within the last month.
Their presence has fuelled rumours that the Gunners have a deal in place, something Crewe boss Steve Davis refuted, but the League 1 manager admitted it's only a matter of time before Clayton moves on.
"I've had no inquires about Max, only people watching, so it's only a rumour. He's not ready to go anywhere," said Davis.
"When the situation is right he will leave to go in somebody's first-team."
Clayton made his debut in April 2010 as a 17-year-old, and has since gone on to make almost 50 appearances for Alex.
Manchester United, who released Michael Owen last summer, signed Clayton's ex-teammate Nick Powell for £4m in the previous transfer window, and the Old Trafford side are reportedly monitoring the progress of his former colleague. But Arsenal, who have a history of signing promising lower-league footballers, most recently Alex Oxlade-Chamberlain, are likely to put up a fight.