The children were admitted at the Baba Raghav Das Hospital in Gorakhpur district of northern Uttar Pradesh, India's most populated state ruled by the right-wing Bharatiya Janata Party.
"The hospital has said that 23 children died on Thursday and 11 today. At the moment we only have this information," senior police superintendent Satyarth Aniruddha Pankaj told AFP by phone.
"The matter is being investigated."
Multiple local media reports said the children died after the company which was supplying oxygen stopped the service, apparently over non-payment of dues running into millions of rupees.
City magistrate V K Srivastava however denied the allegations and said the hospital had enough oxygen cylinders in stock.
"The deaths were not due to lack of oxygen as is being reported. The supply was stopped yesterday (Thursday) but the hospital has enough cylinders in stock," he told AFP.
"Many acute cases are referred to the hospital and on an average 10-11 kids die daily. A team has been set up to look into the matter."
State-run hospitals in India are often stretched to breaking point, with patients facing long delays for even minor treatment and forced to share beds.
As a result, Indians avoid the state-run system if they can, with many flocking to private clinics and hospitals.
But a consultation with a private GP can cost $15, a huge sum for millions living on less than $2 a day.