The girl has been identified as Ruqaya Eid Aboeid from the city of Yatta, sources told The New Arab.
An initial investigation by Israeli authorities said the girl fought with her family before leaving home with a knife.
Reportedly she was 'planning to die', said Israeli police spokesperson Luba Samri.
Ruqaya's father has been arrested and taken in for questioning to find out if the family knew she was planning a knife-attack and if so, if they had tried to stop her.
The use of force by Israeli security forces is subject to heated international debate after Sweden's foreign minister said Israeli forces may be guilty of extrajudicial killings on 12 January.
The comments were condemned by Israeli Prime Minister Benyamin Netanyahu.
At least 164 Palestinians have been killed by Israeli fire since October, 32 of them under the age of 18.
Israel says a Palestinian campaign of incitement is fanning the violence.
The Palestinians say it is rooted in frustration stemming from nearly five decades of Israeli occupation.