Police have received "positive information" that suggests the family of terminally ill Ashya King may now be in Spain.
The family may be travelling to the Marbella area where they have strong links, Hampshire Police added.
The need to find Ashya is now desperate. It is really important that we find him and ensure he receives medical attention at the earliest opportunity.
We still don’t know whether the King family have any spare batteries for the machine which administers food, the knowledge, or any way of recharging the battery.
Without properly administered food Ashya’s situation is very serious.
Ashya's family took him from Southampton General Hospital at around 2pm on Thursday and travelled on a ferry to France some two hours later.
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.