The reported stabbings are the latest in a wave of recent clashes as tensions seethe over a disputed holy site in Jerusalem's Old City.
In one of the incidents Monday, two Israelis, ages 16 and 20, were stabbed in Pisgat Zeev, in the northern part of east Jerusalem. One of the injured, a boy who was riding a bicycle at the time of the attack, is in critical condition, Israeli police spokesman Micky Rosenfeld said. The second person has serious wounds, he said.
Two attackers ran from the scene, and one -- a 17-year-old -- was shot by police as he ran toward them with a knife, Rosenfeld said.
Another stabbing attack in Jerusalem near the police headquarters ended with a suspect shot at the scene, Rosenfeld posted on his official Twitter account. Rosenfeld said the attacker was a woman and one officer was slightly injured.
A Palestinian witness who was walking by heard the gunshots.
"I was walking home when I saw a number of school students that were on their way home from school followed by six or seven settlers who were going after them and provoking them," said the person, who did not want to be named for security reasons. "Then, suddenly I heard shots and saw one of the girls injured on the ground. I don't know if it's the settlers or the police officers who fired towards them."
Attacks on police, soldier reported
Near one of the gates to Jerusalem's Old City
, Israeli police said they shot and killed a young Palestinian man who attacked one of their border officers with a knife.
Border police officers were suspicious of a man walking with his hands in his pockets and asked him to stop and take them out, Israeli police spokeswoman Luba Samri said.
As the officers approached him, the man, a Palestinian, took out a knife and stabbed one of them in his flak jacket, Samri said. Border police responded by shooting and killing the man, she said, without providing details on his identity.
Close friends of the Palestinian's family identified him as Mustafa al Khateeb, an 18-year-old from east Jerusalem who was in his last year of high school.
The police officer was unharmed in the clash, Samri said.
A fourth stabbing occurred on Monday evening, according to Rosenfeld, who said a Palestinian man attempted to grab a weapon from an Israeli soldier inside a bus at an entrance to Jerusalem.
After failing to get the weapon, the attacker attempted to strangle the soldier and stabbed him, wounding the soldier lightly, Rosenfeld said. A passenger aboard the bus struggled with the attacker and a police officer boarded the bus and fought the attacker, who appeared to have grabbed the policeman's weapon, police said.
When a second police officer got onto the bus and saw the attacker with a gun, the officer shot and killed him, Rosenfeld said.
Violence spreads to Gaza
The violence Monday followed a weekend of deadly clashes
and an order by Israeli Prime Minister Benjamin Netanyahu for police reinforcements in Jerusalem.
The Israeli Air Force bombed two Hamas weapon manufacturing facilities in northern Gaza early Sunday in response to a rocket fired into southern Israel.
The exchange of fire suggested the Israeli-Palestinian tensions were spreading beyond Jerusalem and the West Bank.
The Israeli strike on Gaza City caused a house to collapse, resulting in the deaths of a 3-year-old and a 35-year-old woman who was five months pregnant, the Gaza City Fire Department said.
Israel's Iron Dome missile defense system intercepted the rocket from Gaza with no injuries or damage reported, according to the Israeli military. Another rocket launched from Gaza on Sunday hit an open area in southern Israel without causing any reported injuries, it said.
Mounting death toll
Two young Palestinian men carried out knife attacks near Jerusalem's Old City on Saturday in which several police officers and other Israelis were hurt, according to police. Both the attackers were killed by police, authorities said.
Six other Palestinians were killed in clashes with Israeli security forces over the weekend, according to the Palestinian Red Crescent. They included a 13-year-old boy who was hit by a rubber-coated bullet in the West Bank on Sunday, according to the Palestinian Health Ministry.
Hundreds of other Palestinians were wounded in the clashes in Jerusalem, the West Bank and Gaza, the Red Crescent said.
The Health Ministry reported Sunday that 24 Palestinians have been killed since the beginning of October and more than 1,300 have been wounded by live and rubber-coated bullets.
Four Israelis have been killed and several others wounded in knife and gun attacks by Palestinians in Jerusalem and the West Bank since October 1.
Some have suggested the violence represents the start of the third intifada, or uprising, by Palestinians. But others have dismissed that label, saying the unrest is simply the consequence of the absence of any move toward peace.
"We've tried negotiations and it didn't work. So now we will fight," one Palestinian youth in the West Bank city of Hebron told CNN as thick smoke rose from flaming tires.
Tensions over holy site
Amid the continuing attacks, about 1,600 reserve border police officers have been mobilized in Jerusalem and throughout Israel, the Prime Minister's office said over the weekend.
The additional force will continue as necessary as "a primary preventive and deterrent measure," the statement said.
"We are in the midst of a wave of terrorism originating from systematic and mendacious incitement regarding the Temple Mount -- incitement by Hamas, the Palestinian Authority and the Islamic Movement in Israel," Netanyahu said.
The Temple Mount is the Old City holy site that Muslims call the Noble Sanctuary.
Palestinians have repeatedly clashed with Israeli security forces at the site in recent weeks, prompting Israel to restrict access to the site.
Palestinian leaders have suggested the Israeli government is planning to change the status quo at the site, where Jews are allowed to visit but not pray.
Netanyahu has denied the allegations and called on both sides to abstain from going to the site to avoid escalating the situation.