Ashya King was taken out of Southampton General Hospital by his mother and father against medical advice.
Hampshire Police believe he may have been taken to Spain and police are at a Marbella property owned by the family.
Officers said they now thought his family had "taken steps to be able to feed him".
It was unclear precisely what they meant but Hampshire Assistant Chief Constable Chris Shead added: "The feeding system that Ashya needs and the other associated medical care is complex, and I would appeal to Ashya's family not to think that they are able to administer this care themselves."
It is feared the battery on his medical feeding unit, which is designed only for temporary use and is not easy to replace, has now run out.
He said it was unclear whether Ashya's parents had spare batteries.
"Without properly administered food Ashya's situation is very serious," he said.
Ashya and his family were last seen on a ferry to France on Thursday evening.
Mr Shead cautioned it was possible the family may not have travelled on to Spain and asked people across Europe to remain vigilant.
According to Ian Pople, a consultant neurosurgeon, the battery in the feeding machine used by Ashya can't be changed easily.
It is integrated within the machine, much like an iPhone, and it means the machine has to be taken apart to replace the battery.
It's also not designed to be run on batteries for a long period and is usually plugged into the mains.
In other words, it's only battery-reliant for short periods, such as going to the toilet, or moving between wards.
Although the arrest warrant was "based around neglect", Mr Shead said this did not necessarily mean the parents would be charged with that offence.
"It purely gives us the power to arrest and then we will be able to speak to them," he said.
Ashya's paternal grandmother, Patricia King, said the couple were "wonderful" and had been distraught at their child's illness.
Ms King, who lives in Southsea, said her son was "the most caring and wonderful father you could ever have".
She also praised her daughter-in-law, saying she had kept a bedside vigil while Ashya was in hospital.
"We are a very close family," she said.
Mr Shead said his force was considering sending officers from Hampshire to assist Spanish police in Marbella.
He also thanked the Spanish authorities for their help with the search for the family and urged any medical professionals to be on the lookout for the family.
Mr Shead said the latest information was "positive", adding: "There have been widespread media alerts across Europe.
"We would now encourage anyone with links to Spain particularly, to also help us spread the appeal via social media."
Interpol has issued a missing persons alert to all 190 of its member countries and said it was treating all information on the case as "high priority".
Brett King, 51, and Naghemeh King, 45, removed Ashya from the ward on Thursday.
A spokesman for the hospital, which contacted police six hours after the family left, said on Friday: "Ashya was a long-term patient who was permitted to leave the ward under the supervision of his parents as part of his ongoing rehabilitation.
"When the length of time he had been absent became a cause of concern to staff yesterday afternoon they contacted police after a search of the site and attempts to contact the family were unsuccessful."
Mr Shead said police were told by the hospital the child was missing at 20:35 BST on Thursday - more than six hours after he had been taken by his parents.
On why the hospital did not alert police sooner, he said: "That is something that we need to look at."
Parents have the right to remove their children from hospital unless they are prevented from doing so by a court order - it has not been confirmed whether Ashya was subject to an order.
If doctors are concerned that parents intend to remove a child, deny it the medical treatment it needs, and expose it to the risk of serious harm, they can seek a court order.
This will normally involve CAFCASS (the Children and Family Court Advisory and Support Service), the non-departmental public body which provides guardians to represent the best interests of children in family court proceedings.
Once an order is in place, any parent who removes their child in breach of the order is committing a contempt of court for which they could be imprisoned.
Hampshire Police confirmed the family are Jehovah's Witnesses, but there is no suggestion this is why he was taken.
Officers were keeping an "open mind" on the motives, Mr Shead said.
The Office of Public Information for Jehovah's Witnesses said in a statement: "There is absolutely no indication, as far as we are aware, that their decision is in any way motivated by any religious convictions."
Jehovah's Witnesses say they refuse blood transfusions on the basis of biblical teaching. Their website says the Old and New Testaments "clearly command us to abstain from blood".
The family, from Southsea, Hampshire, were travelling in a grey Hyundai I800 Style CRDI, registration KP60 HWK. Officers have asked for anyone who sees the vehicle to contact them.
A European arrest warrant has been issued for the parents of a five-year-old boy with a brain tumour, missing from hospital since Thursday.