Dr Ashraf al-Qudra said that Israeli forces had surrounded the area around the al-Shifa hospital and two others nearby, the Rantisi and al-Nasr.
Speaking to the BBC, he claimed that five strikes had hit the al-Shifa hospital on Friday morning, damaging the maternity ward and other parts of the complex.
Thousands of people are currently sheltering in the hospital, which is the largest in Gaza. Videos verified by several news outlets showed scenes of panic and people covered in blood.
The Israeli army says Hamas hides in and under hospitals and that it has set up its main command center in and under the al-Shifa — claims the militant group and hospital staff deny.
“If the al-Shifa hospital goes out of service it will be a disaster for the people of Gaza City,” Dr al-Qudra said.
Israel Defense Forces (IDF) spokesman Richard Hecht said the IDF does not fire on hospitals but told reporters that Israeli forces would "do what we need to" if Hamas are firing from hospital grounds.
He added: "We’re aware of the sensitivity. If we see Hamas terrorists - we’ll kill them.
"We’re not dropping bombs right now on al-Shifa or Rantisi."
Margaret Harris, a spokesperson for the World Health Organisation, on Friday said that 20 hospitals in Gaza were out of action and that there was “intense violence” at the al-Shifa.
Their claims could not be independently verified.
Earlier on Friday, the Health Ministry said that one person had been killed at the al-Shifa and several were wounded.
The International Committee of the Red Cross (ICRC) warned that the healthcare system in Gaza has “reached a point of no return” amid rising violence.
William Schomburg, head of ICRC sub-delegation in Gaza, said: "The destruction affecting hospitals in Gaza is becoming unbearable and needs to stop. The lives of thousands of civilians, patients and medical staff are at risk."
Israel has been pounding Gaza from the air and sea and on the ground since Hamas gunmen broke through the enclave's border fence on October 7 and carried out an attack in which 1,400 people were killed and about 240 abducted.
The number of Palestinians killed in the war passed 11,000 on Friday, including more than 4,300 children, according to Gaza’s Health Ministry.
The fighting has triggered a humanitarian crisis in the densely populated enclave. On Thursday, a UN World Food Programme official warned that all of Gaza's 2.3 million inhabitants lack sufficient food and face malnutrition.
US Secretary of State Anthony Blinken on Friday said that Israel needs to do more to protect Gazans in his strongest comments to date on civilian casualties.
Speaking to reporters in New Delhi, he welcomed the four-hour humanitarian Israeli pauses the White House announced on Thursday but said further action was required to protect Gaza's civilians.
“Far too many Palestinians have been killed; far too many have suffered these past weeks.
“And we want to do everything possible to prevent harm to them and to maximize the assistance that gets to them,” he said, adding that Washington would be discussing further steps with Israel to advance these objectives.
His comments came as Israel agreed to put in place four-hour daily humanitarian pauses in its assault on Hamas in northern Gaza.
The pauses would allow people to get out of harm's way and for deliveries of humanitarian aid and could be used as a way to get hostages out of Gaza, the White House said on Thursday.