Israeli tanks have reportedly encircled Gaza's Indonesian Hospital, where the Hamas-run health ministry said 12 people were killed on Sunday night.
The hospital's director, Dr Marwan Al-Sultan, told the BBC that the post-operative care department was hit and troops were only about 20m (66ft) away.
On Monday evening, he said intermittent shooting could still be heard on site.
The Israeli military said its forces targeted "terrorists" who had opened fire at them from within the hospital.
The head of the World Health Organization (WHO) said he was "appalled".
"Health workers and civilians should never have to be exposed to such horror, and especially while inside a hospital," Dr Tedros Adhanom Ghebreyesus, wrote on X, formerly known as Twitter.
Health ministry spokesman Ashraf al-Qudra accused Israel of "tightening its noose" around the hospital in the north of the enclave, and later told AFP news agency 200 patients had been evacuated by bus to Nasser Hospital in Khan Younis.
He said the International Committee of the Red Cross (ICRC) was helping to coordinate the evacuation of the remaining 400 patients.
Meanwhile, 28 premature Palestinian babies evacuated on Sunday from Gaza City's besieged Al-Shifa Hospital, which Israeli forces raided last week, have been transported to Egypt for treatment.
Israel launched a major military campaign in Gaza in response to a cross-border attack by hundreds of Hamas gunmen on 7 October, in which at least 1,200 people were killed and about 240 others taken hostage.
Gaza's health ministry says at least 13,000 people have been killed in the territory since Israel launched its retaliatory campaign against Hamas.
The UN Security Council has called for "urgent and extended humanitarian pauses" for "a sufficient number of days" to allow UN agencies to safely enter the sealed-off territory.
But five days on, the International Rescue Committee says the killing and suffering of Palestinians has increased, and has urged the Security Council and others to bring about a ceasefire "without limits on its duration, to protect lives and allow aid to flow".
On Monday morning, video footage verified by the BBC showed several tanks stationed on a road about 240m (800ft) north-east of the Indonesian Hospital, which is funded by Indonesia.
The official Palestinian news agency, Wafa, cited medical sources as saying that dozens of Israeli armoured vehicles were in the surrounding area and that snipers were on rooftops, preventing ambulances from reaching the hospital.
The health ministry said 12 wounded patients and people with them were killed and dozens more were wounded when an Israeli shell reportedly hit the second floor of the facility.
Dr Al-Sultan told the BBC that he heard gunfire all around the building overnight and that at least 10 people were killed.
Al Jazeera posted a video that the Qatar-based network said showed journalist Anas al-Sharif walking through the hospital following the incident. The body of at least one person could be seen, as well as damage to ceilings and equipment in several rooms and corridors.
Sharif reported that "victims are piling up on the floor" and that there was "an overwhelming state of panic among patients".
Hospitals are specifically protected under international humanitarian law. Any military operation around hospitals must take steps to spare patients, medical staff and other civilians inside them.
A spokesman for the Israel Defense Forces (IDF) said: "Overnight, terrorists opened fire from within the Indonesian Hospital in Gaza toward IDF troops operating outside the hospital. In response, IDF troops directly targeted the specific source of enemy fire. No shells were fired toward the hospital."
"Despite the challenges the IDF faces in a war against a terrorist organisation which operates out of hospitals, the IDF is committed to international law and takes numerous measures to minimize harm to non-combatants."
Meanwhile, US President Joe Biden said he believed a deal to free hostages in Gaza was close, as the ICRC said its president had travelled to Qatar to meet with Hamas and Qatari government representatives.
US national security spokesman John Kirby said he believed "we're closer than we've ever been, so we're hopeful".
The ICRC does not take part in negotiations leading to hostage releases but is there to facilitate any deal that is agreed.
That has been the process with the four hostages freed so far - Qatari negotiations with Hamas followed by the ICRC retrieving the hostages from agreed locations and bringing them back to Israel.
Qatar's prime minister has said a deal to free some of the hostages now hinges on "minor" practical issues.
On Sunday night, the IDF released new videos from Al-Shifa Hospital, where it has claimed there was a vast underground Hamas command centre.
It said one showed a tunnel leading to a blast-proof door and that two others taken from CCTV cameras showed two hostages - one of them wounded - being taken to the hospital on 7 October.
The IDF also said a pathologist's report and its intelligence had shown a female Israeli soldier whose body was recovered from a building near Al-Shifa last week had been injured in an Israeli air strike and then killed by a member of Hamas inside the hospital.
Hamas has denied using Al-Shifa and other hospitals as shields for its fighters.
At the same time, the WHO has said it was finalising its plans to complete the evacuation of between 250 to 260 seriously wounded or ill patients trapped at Al-Shifa.
"These include around 29 patients with spinal injuries, 22 patients with kidney failure requiring kidney dialysis," the WHO's Regional Emergency Director, Dr Richard Brennan, told the BBC.
"The vast majority of the remaining patients have complicated war injuries - terrible fractures and amputations, bad burns, head, abdominal, chest injuries, and many of which are complicated by severe wound infection."
Thirty-one premature babies were evacuated from Al-Shifa to the Emirates Hospital in southern Gaza on Sunday by the Palestinian Red Crescent and UN, with the assistance of the IDF. At least five other babies had previously died after being removed from their incubators due to a lack of power after the hospital ran out of fuel.
On Monday, 28 of the evacuated babies were taken to Egypt via the Rafah border crossing for treatment. Some are being admitted to the hospital in El-Arish while others in more serious conditions are being flown to Cairo.
Dr Mohamed Salama, head of the neo-natal unit at al-Ahli Emirates hospital, told the BBC that the parents of two babies had refused to send them to Egypt "due to personal circumstances". A third baby was being kept in Gaza because his condition was stable and his parents were unknown, he said.
Dr Brennan described the evacuation as "one of the few bits of good news we've had during this conflict", but noted that the babies had "a long road ahead of them" to recover.
In a separate development, medical charity Médecins Sans Frontières (MSF) said its clinic in Gaza City came under fire on Monday morning.
"Our colleagues saw that a wall was torn down and part of the building was engulfed by fire as heavy fighting took place all around it. An Israeli tank was seen in the street," a statement said.
An MSF member of staff and 20 family members were in the clinic and in extreme danger, while 50 other people were in nearby buildings, it added.
There was no immediate comment from the IDF.