Mr Crompton was suspended on April 27 because of concerns about public trust and confidence in the force following verdicts in the Hillsborough inquests.
Dr Alan Billings, South Yorkshire Police and Crime Commissioner, said last month that his decision was based on “the erosion of public trust and confidence”.
The force has now begun the statutory process to decide whether to call for Mr Crompton to retire or resign, Dr Billings said.
The first stage of the process is for the crime commissioner to make “a proposal to call for the retirement or resignation of the Chief Constable”.
According to the PCC’s website: “The PCC must obtain the views of the Chief Inspector of Constabulary on the proposal. The Chief Inspector provides his views in writing, and the PCC must take those views into account.”
Dr Billings will now decide whether or not to pursue the proposal to call for retirement or resignation.
Mr Crompton reiterated an apology for the disaster hours after the Hillsborough verdicts last month and said he accepted the findings of the inquest.
But he was criticised by the families of Hillsborough victims for the stance taken by lawyers representing the police during the two-year-long hearings.
He was due to retire in November after leading the force for four years.