The children's hospitals in Dublin - Temple Street, Crumlin, Tallaght and the urgent care centre at Connolly Hospital - say they are at critical capacity.
Children's Health Ireland, which operates them, said there is a perfect storm and the hospitals are under extreme pressure.
High levels of viruses in the community like Respiratory Syncytial Virus (RSV) and flu are contributing to the problem.
Dr Ike Okafor, Clinical Director of CHI, said that due to earlier appropriate measures to protect from Covid-19, families have not experienced illnesses in their children who were born during lockdown.
As a result, hospitals are seeing some very sick babies who need acute care.
CHI is urging families to seek alternative options such as a GP or pharmacy and to make sure their children get the nasal flu vaccine.
But if parents are worried their child is seriously ill or injured, they should attend their nearest paediatric emergency department.
Meanwhile, all routine procedures at Belfast's paediatric hospital have been postponed after an increase in children presenting with bacterial and viral infections.
In a statement, the Belfast Trust said the hospital is experiencing "very significant pressures".
It said 227 children attended its emergency department yesterday, and that the number of children attending with symptoms of bacterial and viral infections has increased.
"Unfortunately, we have taken the very difficult decision to postpone all routine procedures at the Royal Belfast Hospital for Sick Children to allow our staff to care for those children with serious or time critical illness at this time.
"We appreciate how difficult this will be for families and children and we apologise for the distress this may cause."