Merseyside Police say they are considering Mr Anderson's letter requesting an investigation.
In the summer, Barkley was set to leave Everton for Chelsea for £35m but pulled out of the deal at the last minute citing fitness concerns.
Chelsea eventually signed him up for a much lower £15m this month, amid a campaign where the 24-year-old hasn't played a single minute of competitive football.
Self-confessed Everton fan and Liverpool mayor Anderson now wants an inspection of the deal, and has also written to FA chairman Greg Clarke and Premier League boss Richard Scudamore to lay out his concerns.
Anderson wrote: “(This is a) decrease in value of more than one million pounds a week. At best, this represents a very poor deal for Everton Football Club.
“At worst, it could be seen as a deliberate attempt to drive down a player’s value in the transfer market so as to benefit the player, his agent and the buying club.
"There seems to be at least a public perception that collusion has taken place.”
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