The source said an 18-year-old member of Islamic Jihad was killed and several other people wounded in the gun battle with Israeli forces early on Tuesday.
Palestinian officials also told Reuters that a civilian had been killed during the Israeli incursion into the town of Jenin and surrounding villages in the West Bank.
However, an Israeli army spokesman said soldiers had shot an armed man in the village of Qabatiya, near Jenin, after armed men opened fire and threw explosive devices at troops. Medics named the dead man as Mohamad Zakarneh.
An army spokesman said: "Forces identified two armed men and opened fire on them. One of them was hit."
Meanwhile, Palestinians said undercover soldiers opened fire on a car in the West Bank city of Nablus, killing a local leader of the Fatah-linked Al Aqsa Martyrs' Brigades and wounding his cousin.
The Israeli military said the slain man was responsible for attacks against Israelis.
Hamas fighter slain
In other incidents, a fighter from the armed wing of Hamas and another Palestinian were killed by the Israeli army during an incursion into the north of the Gaza Strip, Palestinian hospital officials said on Tuesday.
The dead fighter was Zaher Atamani, 21, of the Izz al-Din al-Qassam Brigades. He was shot near the Jabalya refugee camp.
The other victim was identified as Ghazi Abu Dahruj, 25, but medical officials could not immediately say whether he was an armed fighter or a civilian.
The upsurge in violence comes as Israel steps up military action against Palestinians firing rockets at Israeli towns from the Gaza Strip.
A series of Israeli raids on the Gaza Strip have killed 21 Palestinian, mostly armed fighters, since Thursday.
Also on Tuesday, the Israeli military said its forces discovered a tunnel under the border between Gaza and Egypt near the Israeli border. The military said it was used to smuggle weapons into Gaza.
Soldiers arrested seven Palestinians in raids throughout the West Bank, an Israeli military statement said.
The Israeli army says it is looking into reports that a second Palestinian had also been killed.
An Aljazeera television correspondent reported that more than 50 military vehicles raided the town to arrest suspects.