Emergency services were called to the scene, in Kirkwood Street, Rutherglen, at about 16:15 on Tuesday 23 August.
The two-year-old boy was taken to Glasgow's Queen Elizabeth University Hospital where he is still believed to be in a stable condition.
A police spokeswoman said a 29-year-old man had been reported to the procurator fiscal in connection with the incident.
A man has been reported to prosecutors after a toddler fell from a flat window in South Lanarkshire last week.