Currently, Scotland has two clinics - in Edinburgh and Glasgow - which specialise in the complex surgical procedures.
But Scottish Government plans to centralise operations means surgery will only take place in Glasgow with many children and families facing longer journeys for treatment.
Last year a consultation took place on the proposals, but a number of concerns were raised about the process which many parents described as “flawed” and “lacking transparency”.
And The Cleft Lip and Palate Association (CLAPA) spoke out about the lack of evidence to show that the existing two sites are not working.
If plans go ahead, fears have been raised that it will significantly compromise the medical teams and service, leading to less frequent clinics.
Almost 100 babies are born every year in Scotland with a cleft lip or palate and surgery can help them talk and eat.
Scottish Conservative public health spokesman Miles Briggs said:
“I have written to Shona Robison asking her to intervene and pause any further consideration of these centralisation plans until the many concerns of parents, patients and clinicians across Scotland can be adequately addressed.
“It is alarming that the National Services Division say this is a minor service change when it seems very clear that this is a major change to the provision.
“There are a large number of questions over how this process has been handled and I have now been provided with additional information which was not part of the consultation and which suggests care and audited outcomes for children following surgery in Edinburgh are extremely high.
“Parents feel very strongly that the care and support which the high performing Edinburgh team currently delivers is world class. It is vital that this is not lost along with the specialists.
“I also share the concerns of many parents in Lothian and elsewhere in Eastern Scotland that a centralisation of this surgery would mean longer journeys for care and the associated extra stress on families.”