A spokeswoman said there were only 35 doctors left to care for hundreds of trapped patients, and that the number of casualties was rising.
Medical supplies are also running out, and there is a shortage of blood.
Russian-backed Syrian government forces launched an assault on eastern Aleppo on Thursday after a truce collapsed.
Eastern parts of the city are held by rebels.
Since then, the intense and sustained aerial bombardment of Aleppo and its surrounding countryside has killed at least 248 people, almost all of them civilians, according to the UK-based Syrian Observatory for Human Rights.
The US and its allies, who support the rebels fighting the Syrian government, have said the attacks constitute war crimes and accused Russia of "barbarism". Moscow has vigorously denied the allegations and criticised their "unacceptable" rhetoric.
Aleppo, once Syria's largest city and the country's commercial and industrial hub, has been divided roughly in two since 2012, with President Bashar al-Assad's forces controlling the west and rebel factions the east.
In the past year, government troops have gradually broken the deadlock with the help of Iranian-backed militias and Russian air strikes. Earlier this month, they severed the rebels' last route into the east and placed its 250,000 residents under siege.
At a news conference in Geneva on Monday, WHO spokeswoman Fadela Chaib said the escalation of fighting was "claiming more victims every day" and called for the "immediate establishment of humanitarian routes to evacuate the sick and wounded from the eastern part of the city".
"We are talking about only 35 doctors left in east Aleppo to take care of hundreds of wounded people, and the number is increasing," she warned.
"It's not just one attack - it's a daily attack on civilians. So you can imagine how many wounded are trying to get some medical help in such circumstances," she added. "It's a worrisome situation."
Many streets are now blocked by rubble, meaning ambulances cannot get through.
All of the 25 functioning or partially functioning medical centres are on the verge of complete destruction, according to the WHO.
On Monday, Dr Abd Arrahman Alomar of the Syrian American Medical Society said at least 40 wounded people needed to be evacuated, but that most wanted to be sent to rebel-held territory in neighbouring Idlib province or to Turkey, and not to the government-controlled west of Aleppo.
Both the United Nations and the International Committee of the Red Cross have been calling for humanitarian corridors into Aleppo for several weeks, but so far those calls have been ignored, reports the BBC's Imogen Foulkes in Geneva.
On the ground in Aleppo on Tuesday, government forces were reportedly making advances in the centre of the city, attacking rebel-held districts on four fronts and also mobilising armoured vehicles and tanks.
Military sources and rebels say pro-government forces appear to be gearing up for a possible ground assault after several days of heavy airstrikes against rebel-held areas.
The Britain-based Syrian Observatory for Human Rights confirmed that clashes are taking place in central Aleppo.
A military official earlier told The Associated Press news agency that the offensive will not halt until insurgents are "wiped out".
A senior rebel official told Reuters news agency that fighters had so far been able to repel the assaults around the Palestinian refugee camp at Handarat, to the north, the nearby Kindi hospital area, the central Rashidin district, and the 1070 Apartment Project area in the south-west.