Police said missing Ashya King's health will "deteriorate rapidly" if he is not found today and that his condition would become "life-threatening".
Assistant Chief Constable Chris Shead of Hampshire Constabulary said: "He is being fed through a tube. The feeding system is battery operated and the battery will run out today."
He added police "had every reason to believe" that the King family remains in France, where they travelled to yesterday.
Ashya King's parents told the Daily Mail they fled the UK after being told NHS treatments would leave him with life-long disabilities.
The head of the police force which applied for a European Arrest warrant against the parents of Ashya King has now urged their release.
A British mother whose son's life was saved by proton therapy has told ITV that she defied NHS doctors' advice to pursue the treatment.