The MSF facility is seen as a key medical lifeline in the region and has been running "beyond capacity" during recent fighting that saw the Taliban seize control of the provincial capital for several days.
The strike early Saturday left the building engulfed in flames, with photos posted by MSF on Twitter showing shocked and dazed staff in the aftermath of the bombing.
"At 2:10 am (2040 GMT) local time... The MSF trauma centre in Kunduz was hit several times during sustained bombing and was very badly damaged," it said in a statement.
"Three MSF staff are confirmed dead and more than 30 are unaccounted for. The medical team is working around the clock to do everything possible for the safety of patients and hospital staff."
Kunduz has seen heavy fighting since Taliban insurgents stormed the provincial capital on Monday -- the first major city to be captured by militants since 2001.
MSF said some 105 patients and their caregivers and more than 80 international and local MSF staff were in the hospital at the time of the bombing.
The charity had stated Friday it was treating 59 children at the facility.
NATO said US forces may have been behind the strike.
"US forces conducted an air strike in Kunduz city at 2:15 am (local time)... Against individuals threatening the force," a NATO statement said.
"The strike may have resulted in collateral damage to a nearby medical facility. This incident is under investigation."
A local resident said he was desperately trying to contact six of his friends, all doctors and nurses at MSF, but their cellphones were unreachable.
"I don't have any news from them and they may have been killed," he told AFP.
He said many residents with wounded relatives at the facility were too afraid to step out of their homes to check on them amid erratic firefights between troops and Taliban snipers lurking in the city.
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