Al Jazeera's Ayman Mohyeldin, reporting from the Gaza Strip, said that much of the fighting took place around a high area on the outskirts of the city.
"The strategic overlook essentially gives an overview of the entire northern part of Gaza .. that's why it probably has some significance to the Israeli military," he said.
"Because of its height and elevation it has probably also been used to fire rockets into Israel."
Fierce fighting between Israeli troops and Palestinian fighters was also reported in eastern Jabliya in the northern Gaza Strip.
"All of Gaza is dark except for the flares fired by F-16s and Apaches [helicopter gunships]. The F-16s for the first time are flying very low and shooting missiles everywhere," Moussa el-Hadda, a retired doctor in the Gaza Strip, said.
"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.
Despite 10 days of constant bombardment of the Gaza Strip by the Israeli military, causing the deaths of at least than 548 Palestinians, a senior Hamas official said on Monday that "victory is coming" for Palestinian fighters.
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 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 operation.
But Major Avital Leibovitch, an Israeli military spokeswoman, told Al Jazeera that the cross-border attacks were being slowed.
"As long as the aggression intensifies, your losses will increase and you will sink further into the Gaza quagmire"
Izz-e-din Al-Qassam Brigades spokesman
"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 Izz-e-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."
Al Jazeera's Mohyeldin said that the ongoing fighting was "not a battle between two armies".
"The Palestinian factions' biggest weapon, they say, is their will and resolve. What they are using are small to medium sized machine guns ... nothing particularly sophisticated," he said.
"The style of warfare they are engaging in is a cladestine one where they can simply melt away, they have the agility advantage over the Israelis because they are not operating large vehicles, they are mostly on foot."