"Protecting these children, many of whom are refugees and internally displaced... must be a priority", said the UN's children's agency Unicef.
Despite deals with the Ethiopian government, humanitarian agencies say they are being denied access to Tigray.
Government forces have been battling Tigray fighters since 4 November.
The government says it is in control of the region and the conflict is over. But Tigray People's Liberation Front (TPLF) says it is still fighting on various fronts.
Hundreds, even thousands, of people are thought to have been killed in the conflict, while about 50,000 have fled to neighbouring Sudan.
In a statement, Unicef said: "The longer access to [the children] is delayed, the worse their situation will become as supplies of food, including ready-to-use therapeutic food for the treatment of child malnutrition, medicines, water, fuel and other essentials run low."
It added: "We call for urgent, sustained, unconditional and impartial humanitarian access to all families in need wherever they are."
Neither the Ethiopian government nor the TPLF have commented on the issue.