Steven Chesney, 33, was being held by police in Motherwell earlier this year when he began feeling unwell and later died.
The Police Investigations & Review Commissioner (Pirc) said it had now submitted a report to the Crown Office in connection with the incident.
In a statement, the Pirc said: "The Police Investigations & Review Commissioner (Pirc) has submitted its report to the Crown following an investigation into the death in custody of a 33-year-old man in Motherwell.
"The investigation focused on the circumstances leading up to the death in custody of Steven Chesney on 1 May 2016.
"The PIRC's report has now been submitted to the Crown Office and Procurator Fiscal Service (COPFS) for their consideration."