The Independent Police Complaints Commission (IPCC) said it has received a complaint over Mr Crompton's conduct at the hearings.
Inquests concluded 96 Liverpool fans were unlawfully killed and found police failures contributed to the deaths.
Mr Crompton was suspended after the inquests over "an erosion of trust".
The complaint, from a legal representative of a number of Hillsborough families, was forwarded to the office of South Yorkshire Police and Crime Commissioner Dr Alan Billings for a recording decision before being referred to the IPCC.
The IPCC said it is now "conducting an assessment of all available information to decide how the matter should be progressed".
Mr Crompton was due to retire in November after leading South Yorkshire Police for four years.
He was suspended following the conclusion of the inquests in April, with Dr Billings saying he had "no choice" but to act following an "erosion of trust".
A police watchdog is examining the actions of South Yorkshire Chief Constable David Crompton and his force during the Hillsborough inquests.