A spokesman for Charlie Gard's family has described Great Ormond Street's claim that staff have received death threats as "arguably a cynical ploy".
Charlie's parents have accused hospital staff of overstating the severity of his condition and have frequently clashed with lawyers and the judge during a hearing at the High Court.
Mary MacLeod, chairman of Great Ormond Street Hospital, said that while she appreciated feelings around the ongoing court battle between medical staff and Charlie's parents were running high, nothing could justify the abuse. They are fighting to remove the infant from life support in an attempt to allow him to "die with dignity".
Charlie suffers from a rare genetic condition called mitochondrial DNA depletion syndrome that causes progressive muscle weakness and has left him with brain damage and unable to breathe without a ventilator.
"Despite conflicting issues, we have always had the utmost respect for all the staff who work tirelessly at Great Ormond Street Hospital and the very hard jobs they do every day".
MacLeod said families visiting other ill children have also been "harassed and discomforted" on the grounds of the renowned hospital in London. His parents want to take him to the United States for pioneering treatment. Staff have received abuse both in the street and on line.
In a statement, she said numerous messages were menacing, including death threats, and that the hospital was in close contact with the police.
The Gards, whose case is being followed by US President Donald Trump, were last week granted legal and permanent residence by the US Congress in order to pursue the experimental treatment.
Charlie's parents have lost all previous court cases, including one before the European Court of Human Rights, which were created to force the hospital to let them bring their son to the United States for an experimental treatment. "We fully understand that there is intense public interest, and that emotions run high", she added.
Court is schedule to resume early next week, but despite American lawmakers going so far as to grant the infant and his parents temporary us citizenship, it appears unlikely that the court will allow Charlie to be transported for treatment that has been ruled to have "no prospects of success" and which "would offer no benefit". As The Guardian reports, the Great Ormond Street Hospital has now received thousands of abusive messages from those opposed to removing Charlie Gard from life support.
"We recognise the tireless advocacy of Charlie's loving parents and the natural sympathy people feel with his situation".
On Friday the court heard that a new MRI scan on Charlie made for "sad reading".
The court is due to hold a hearing on Monday to consider the latest medical evidence.