A spokesman for Afghanistan’s interior ministry described those killed as “medical staff”.
It was unclear whether the cop was dead or in custody.
Afghanistan suffered a spate of deadly attacks in the run-up to presidential elections held on April 5, including a restaurant bombing in January and an attack on a hotel in March.
Thursday’s attack took place at a hospital in western Kabul run by CURE, a small US Christian charity.
The US embassy in Kabul tweeted that it was “with great sadness we confirm that three Americans were killed in the attack on CURE Hospital”. No other information would be released for the moment, it added.
Police say that the attack happened when a policeman at a nearby checkpoint opened fire on a group of five or six foreigners who were entering the hospital in western Kabul.
A spokesman for the interior ministry said that three had died, including two women. Others were being treated for their injuries, officials said.
Some reports say that the policeman, who was named as Ayunullah, shot himself after the incident. But the Reuters news agency says that he is wounded and in police custody.
The charity CURE took over the hospital seven years ago and restored it to specialize in providing healthcare for women and children. The 100-bed hospital employs 27 doctors and 64 nurses, CURE says.
It is the second attack on a Christian target in the city this year.
In March, the Taliban attacked a guest house used by foreigners working for a faith-based agricultural charity.
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