The baby was found at Morrisons on the Black Country Route in Bilston, Wolverhampton, on Sunday morning.
Thought to be premature, he was found alone by a member of the public.
West Midlands Police has arrested a 21-year-old woman and 36-year-old man in connection with the baby’s death and they are currently being held in custody.
The force would not confirm what offence the pair were arrested on suspicion of and declined to comment as to whether they were related to the baby.
Officers launched an appeal on Sunday to trace the boy’s mother, who was later identified and taken to hospital to receive “appropriate care”.
“This was a devastating discovery which I know has shocked and saddened many people,” Det Insp Tim Draper said.
The force will carry out a post-mortem examination on the baby to establish his cause of death but in the meantime urged people “not to speculate on the circumstances”.