Fouad Nafaa, head of the Rafidia Hospital in Nablus, said Ms Eygi had arrived at the hospital in a critical condition with a serious head injury.
“We tried to perform a resuscitation operation on her, but unfortunately, she died,” he told Reuters.
On Friday, the White House condemned the death of Ms Eygi, but did not say whether she had been shot by Israeli troops.
Sean Savett, a spokesperson for the National Security Council, said Washington was “deeply disturbed by the tragic death of an American citizen” in the West Bank.
“We have reached out to the government of Israel to ask for more information and request an investigation into the incident,” Mr Savett added in a statement.
Ms Eygi was reportedly born in Antalya, a city in Turkey’s south. Oncu Keceli, a spokesman for the Turkish foreign ministry, said the country would exert “all effort to ensure that those who killed our citizen is brought to justice”.
In a statement, the IDF said that their forces had “responded with fire toward a main instigator of violent activity who hurled rocks at the forces and posed a threat to them”.
It continued: “The IDF is looking into reports that a foreign national was killed as a result of shots fired in the area. The details of the incident and the circumstances in which she was hit are under review.”
In a separate incident in the West Bank village of Qaryut on Friday, a 13-year-old girl was killed during clashes with Israeli forces, Palestinian health officials said.