The Palestinian health ministry identified the victims as Khaled Darwish, 21, Qassam Sariya, 19, and Ahmed Saqr, 15, Qassam Faisal Abu Sirriya, 29.
The fifth victims has yet to be identified.
The ministry added that at least 60 people have been injured. At least 10 of those injured are in critical condition.
Jenin has been a regular target of Israeli assaults in the West Bank over the last year.
At least eight Israeli soldiers are reportedly wounded, some of them seriously, according to Palestinian news site Arab48 and the Israeli media outlet, Haaretz, though this has not been confirmed by Israel.
Early Monday, a large number of Israeli troops stormed the city in the northern West Bank, deploying snipers on some houses, and violent confrontations broke out in several areas, during which soldiers fired live bullets, stun grenades and tear gas, and attack helicopters were used.
The Palestinian Islamic Jihad resistance group claimed its members had detonated several explosive devices near Israeli military vehicles and then targeted them with gunfire.
Footage circulating online show Israeli army vehicles coming under heavy gunfire fire with explosions in the background. Another video shows an Israeli military jeep facing an apparent improvised explosive device.
One clip circulating online shows an Israeli attack helicopter firing at targets in the city.
The Israeli military rarely uses aircraft in its operations in the occupied West Bank. Israeli media reported that it was the first use of an attack helicopter in the territory since the Palestinian uprising in the early 2000s.
An Israeli army spokesperson on Monday said that military forces had entered the city to arrest Palestinian suspects. During the operation, “massive exchanges of fire occurred, and improvised explosive devices were thrown at the forces who responded by firing”, the official added.
Tensions in the West Bank have been on rise over the last year, with Israeli forces carrying out near-nightly raids resulting in clashes with Palestinian resistance groups.
Israeli forces and settlers have killed at least 159 Palestinians this year, including 26 children.
A total of 122 fatalities have been recorded in the West Bank and East Jerusalem and a further 34 in the Gaza Strip.
Palestinians have killed at least 20 Israelis in the same period.