The deaths were the biggest blow to Israel's military since it launched its ground assault on Saturday, with only one other soldier killed in the previous two days.
Nonetheless, witnesses told Al Jazeera that the Israeli ground offensive was closing in on Gaza City, where Israel believes most of the Hamas leadership to be.
Large explosions and intense ground battles close to the centre of the city in the eastern neighbourhood of Shejaiya, were reported late on Monday and into the early hours of Tuesday.
Flares lit up the skies as Israeli attack helicopters and fighter jets flew low over the city.
A spokesman for the al-Quds Brigades, the military wing of the Palestinian Islamic Jihad movement, told Al Jazeera that Israeli tanks were trying to move into Gaza City.
He said that the group's fighters had destroyed an Israeli armoured personnel carrier in the clashes.
Reports early on Tuesday suggested that Israeli naval ships had fired on the coastal town of Deir al-Balah, killing 10, and that the army was moving in on Khan Younis, a town in the south of the Gaza Strip.
Al Jazeera's Ayman Mohyeldin, reporting from Gaza, said that much of the fighting was taking place around a high area on the outskirts of Gaza City.
The area gives an overview of the entire northern part of Gaza, which would be strategically useful to the Israeli military, and because of its height and elevation, it had probably also been used to fire rockets into Israel, our correspondent said.
Fierce fighting between Israeli troops and Palestinian fighters was also reported in eastern Jabaliya in the northern Gaza Strip.
Moussa el-Hadda, a retired doctor in the Gaza Strip, said "all of Gaza is dark except for the flares fired by F-16s and Apache [attack helicopters]".
|Flares lit up Gaza City as Israel used artillery, fighter jets and attack helicopters [AFP]
"The F-16s for the first time are flying very low and shooting missiles everywhere.
"Nobody can leave their home, we have no shelters and Israel knows this. They just bombed everywhere and most of their bombs are hitting houses," he told Al Jazeera.
Earlier, Ehud Barak, the Israeli defence minister, told members of parliament that Gaza City was partially surrounded.
"We have hit Hamas hard, but we have not yet reached all the goals that we have set for ourselves," he said.
Dozens of Palestinians were reported to have been detained during the Israeli push into the Gaza Strip, with Israel saying they were Hamas fighters.
Despite 10 days of constant bombardment of the Gaza Strip by the Israeli military, a senior Hamas official said on Monday that "victory is coming" for Palestinian fighters.
"As long as the aggression intensifies, your losses will increase and you will sink further into the Gaza quagmire"
Izz al-Din al-Qassam Brigades spokesman
Mahmoud al-Zahar said that the movement's armed wing had "given the most beautiful performances during its confrontation with the army that the world thought invincible".
"We will defeat it, God willing," he said.
Israeli officials say the assault on Gaza is aimed at stopping Hamas from firing rockets into southern Israel.
Al-Zahar warned that Israel's war on Gaza had opened it up to retribution by Palestinian fighters.
"They have legitimised the murder of their own children by killing the children of Palestine," he said.
"They have legitimised the destruction of their synagogues and their schools by hitting our mosques and our schools."
Palestinian fighters have continued to fire rockets into southern Israel despite the Israeli military offensive.
But Major Avital Leibovitch, an Israeli military spokeswoman, told Al Jazeera that the cross-border attacks were being slowed.
"Those rockets are coming less and less, because today we have had approximately 20 launchings which is one-tenth of Hamas's original launching capabilities ... there are signs that Hamas is weakening," she said.
"The second stage of the operation is going as planned, we have a variety of forces joining in this and it is proceeding according to the plan."
Abu Obeida, the spokesman for Hamas's military wing, the Izz al-Din al-Qassam Brigades, warned the Israeli military that it had many fighters ready to face it in the Gaza Strip.
"We have prepared thousands of brave fighters who are waiting for you in each corner of the street and will welcome you with fire and iron," he said.
"As long as the aggression intensifies, your losses will increase and you will sink further into the Gaza quagmire."