Liverpool mayor Joe Anderson wrote to Merseyside police this week, asking them to investigate the transfer.
Chelsea completed the signing of Barkley on a five-and-a-half-year deal on Friday.
But just three months previously, the England midfielder failed to complete a £35m switch despite undergoing a medical at Stamford Bridge.
The reasons behind the decision were unclear, but it’s thought recovery from his hamstring injury played a part.
Anderson speculated as to whether there was “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”.
Barkley, 24, had been at Everton since the age of 13.