A senior judge has raised concern about people in the position of Alfie Evans's parents being vulnerable to getting bad advice.
Lord Justice McFarlane raised concern at a Court of Appeal hearing on Wednesday.
He said there was evidence that Tom Evans and Kate James had made decisions based on incorrect advice.
The judge, who headed a panel of appeal judges considering Alfie's case, said similar issues had arisen in other recent cases.
He said there was a "darker side" to some offers of support and suggested that some form of investigation should be staged.
It comes as Merseyside Police confirmed they were monitoring social media posts commenting on Alder Hey Hospital, where the toddler is being treated.
Alfie Evans has an unknown and undiagnosed brain condition.
His parents have fought to take him to Italy for treatment - but have failed in an 11th-hour attempt to persuade judges to let them do so.
Its the latest in a long running legal dispute over Alfie's care. Medics have argued that further treatment was futile and not in Alfie's best interests.
The High Court has ruled with doctors that life support could be withdrawn - neither the Appeal nor the Supreme Court have overturned that decision.