In one incident a Palestinian stabbed and lightly wounded an Israeli border guard near a shrine known to Jews as the Cave of the Patriarchs and to Muslims as the Ibrahimi Mosque, in the volatile city of Hebron.
"A Palestinian attacked and stabbed a soldier. Security forces responded and shot the attacker. The soldier sustained light injuries," the army said.
Police spokesman Micky Rosenfeld confirmed that the attacker, aged 24, had died.
"Moments ago forces thwarted the second attempted stabbing today in Hebron. The Palestinian assailant attempted to stab an IDF soldier at a security post," read an army statement.
Police spokeswoman Luba Samri said the attacker had been shot dead.
Both incidents were followed by clashes between young Palestinians and Israeli security forces, according to an AFP journalist on the scene.
A wave of such knife attacks and shootings has left nine Israelis dead since early October.
One Israeli Arab attacker has also been shot dead.
While a spate of protests and attacks in Jerusalem has eased, tensions have flared in Hebron, where near-daily clashes pit youths against Israeli soldiers enforcing the decades-long occupation of the West Bank.
Protests in recent days have erupted over Israel's policy of withholding the bodies of attackers, one of a series of measures to try to dissuade attacks on Jews, which began in early October as tensions over the Al-Aqsa mosque compound in annexed east Jerusalem boiled over.
Israeli Prime Minister Benjamin Netanyahu has repeatedly denied seeking to allow Jews to pray at the compound, which they refer to as the Temple Mount.
Only Muslims are allowed to pray within the compound, while non-Muslims can visit but not pray there.