Police were tipped off about a plot to kill Mangal, a government spokesman, said.
Speaking from Milan, Cecilia Strada, the head of Emergency, told Al Jazeera that this was a completely groundless claim.
"It sounds quite ridiculous that they would be involved in any plot. They have spent the last few years in Afghanistan, helping and treating people for free.
"We ask that you respect their rights, first of all, the right to communicate with us and let us know where they are and what their condition is."
The Associated Press obtained a video of the raid that shows British troops accompanying Afghan police, soldiers and government officials to the hospital.
In a storeroom, boxes are opened containing what appear to be bullets, pistols, hand grenades, and bags of explosives.
A British soldier is heard saying that an explosives-disposal unit was on its way when the explosives were found.
The three Italians are then shown sitting on outdoor benches but the names on their identification cards are not visible.
|Three Italians and six Afghans were arrested for allegedly plotting to kill governor [AP]
Emergency said it had been unable to make contact with its employees since their arrest.
"The only contact we have been able to make has been through one of the employee's cell phones answered by someone who identified himself as a British military official," Strada, Emergency's head, said.
A Nato spokesman in Kabul said its forces had not taken part in any arrests.
Franco Frattini, the Italian foreign minister, was closely following developments, a foreign ministry spokesman said.
"Pending details in this matter, the government reaffirms its strictly rigorous line against any direct or indirect support for terrorism, be it in Afghanistan or elsewhere," the spokesman said.
Emergency has had a tense relationship with local authorities due to its policy of treating all patients, including suspected members of the Taliban.
It has operated in Afghanistan since 1999 and currently runs three surgical centres, a maternity centre and a network of 28 health centres that they say have treated 2.5 million people.
In another security-related development, Hamid Karzai, the Afghan president, was forced to cut short a trip to Kunduz, a province in the country's north, after a rocket attack.
He had been travelling with US General Stanley McChrystal to meet tribal elders and German troops when rockets fell near a base he was due to visit on Saturday.
Al Jazeera's James Bays, who was covering the Kunduz visit when the incident occurred, said the security situation in the area is volatile.
"Kunduz is the most violent part of the north of Afghanistan," he said. "He [Karzai] is trying to win over some of those elders who in the past might have supported his enemies."
The joint visit with McChrystal came as the White House tried to move on from a public row between Karzai and the US.
The dispute was sparked by Karzai's comments blaming the fraud in last summer's presidential election on foreign embassies in the country.