Protesters stationed outside the hospital where Alfie Evans is being kept on life-support have been accused of intimidating and verbally abusing other parents and staff.
Alfies parents have now apologised after the abuse from some of the hundreds of supporters gathered outside Alder Hey Childrens Hospital in Liverpool emerged.
Merseyside Police said they were investigating reports of instances of verbal abuse and acts of intimidation.
Tom Evans, 21, and Kate James, 20, want to move their 23-month-old son who is said to be in a semi-vegetativ state to a hospital in Rome for treatment.
They already lost fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
Yesterday, Mr Evans said in a statement on Facebook: Deepest apologies from me and Kate James to the parents and staff effected by the protest/demonstration,.
It wasnt mine Kates or anyone else intention to harm or cause conflict or upset.
We just wanted to take our son to give him the chance he deserves.
Chief Inspector Chris Gibson said officers recognise the sensitivities involved in this very difficult and sad situation.
Whilst many people have gathered to protest in a peaceful way, Merseyside Police is now investigating a small number of reports, some of which originate from social media, as well as instances of verbal abuse and acts of intimidation from those outside the hospital, he said.
This is extremely unhelpful for all concerned and we are investigating further to establish the full circumstances.
We would like to remind the public that this is a hospital for sick children and it should not be forgotten that many families are going through extremely challenging and emotional times.
We would ask protesters to respect families and staff, including the poorly children in the wards and to ensure that access to the hospital is not restricted at any time, so that services including the blood and ambulance service can run as efficiently as possible.
He urged any victims of intimidation or harassment to report it to the police or hospital.
Alfies parents have already lost fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
On Monday Court of Appeal judges Lord Justice Davis, Lady Justice King and Lord Justice Moylan ruled against the parents in London.
They also rejected a request for the case to go back to the Supreme Court.
Mr Evans emerged from the hospital two hours after the decision was given by the court in London.
Wiping away tears and visibly shaking, he thanked supporters surrounding him and again asked for his sons life not to be terminated and for other doctors to be allowed to look after him.
He added: They cant break us, we are never going to back down.
Alfie, the family and all of our supporters are stronger than ever and we will keep fighting all the way.
We will never give up on you Alfie.
Judges have heard that Alfie, born on May 9 2016, has a degenerative neurological condition doctors had not definitively diagnosed.
At the appeal hearing, Lord Justice Davis said doctors had agreed that there was no hope for Alfie.
In statement Alder Hey said it refuted criticisms from Mr Evans of the treatment Alfie has received.
The statement adds: Our priority is to continue to provide the best possible care for Alfie and his parents at this difficult time.