The study warned that a repeat of the closure of the paediatric ward at St John's Hospital, Livingston – which shut for three weeks last summer – was highly likely.
The unit at Borders General Hospital was also identified as at risk of closure, and concerns were raised about specialist staffing levels at facilities throughout Lothian, Borders and Fife health boards.
The report, by NHS Scotland's South East and Tayside regional planning group, follows a £500,000 rescue package by the Scottish Government to resolve problems at St John's.
It warned: "The ongoing difficulties with the level of gaps in the middle grade paediatric medical trainee workforce will continue and there is a high risk the circumstances that led to the temporary closure of the inpatient paediatric services at St John's Hospital will recur unless rota gaps left by the withdrawal of trainee cover out of hours can be filled with alternative arrangements.
"Should the level of gaps worsen then services at Borders General Hospital would also be at risk. The current workforce model is therefore unsustainable."
A Scottish Government spokesman said: "We are aware of the situation in the south-east of Scotland and have made clear to the three NHS boards involved that we will support every effort to ensure the best quality paediatric services are maintained."