The police say they have arrested eight people who were behind the event, including the main organiser, Naomi Silekunola, a well-known figure in the city.
President Bola Tinubu has extended his heartfelt condolences to the parents who have lost their children, and ordered a thorough investigation.
The Oyo state government said victims had been taken to hospitals across Ibadan for treatment after the crush at the Islamic High School in the city’s Bashorun district.
Authorities have urged parents who are concerned about the whereabouts of their children to check at the city’s hospitals.
Medics in one hospital told the BBC that six children had been admitted but only two survived – four had died.
At another hospital, a doctor who declined to be named said he had counted three dead bodies.
Some affected parents told journalists that they had accompanied their children to converge at the venue of the “End of the Year Christmas funfair” as early as 05:00 on Wednesday – five hours before the event was due to start.
They were hoping to get some money and food as the organisers had promised to give out 5,000 naira each (about £2.50; $3) to 5,000 children as well as make free food available to attendees.
Nigeria is grappling with its worst economic crisis in a generation, which explains why more than 10,000 people reportedly turned up for the event.
There have been several similar incidents this year.
In March, two female students were crushed to death at the Nasarawa State University, Keffi, near the capital Abuja, when a rice distribution programme by the state governor caused a crowd surge. At least 23 people were injured.
Three days later in the northern state of Bauchi, at least seven people died in another crush when a philanthropist and businessman was giving handouts of 5,000 naira.
Earlier in February, five people were reported killed in Lagos when the Nigerian Customs Service auctioned seized bags of rice.
A crowd surge for bags of rice being auctioned for about $7:00 led to the trampling to death of five people with dozens more injured.
