A government compound was also hit.
Israel has made no comment on the latest strikes other than to say they will press ahead with the campaign in the face of mounting international criticism.
Earlier on Sunday Israeli aircraft bombed the length of the Gaza-Egypt border, taking out tunnels used to smuggle in vital goods to the besieged strip.
Dozens of tunnels are said to criss-cross between southern Gaza and Egypt's Sinai desert, providing a lifeline to residents who are starved of basic supplies due to an 18-month-long Israeli blockade.
Avital Leibovitch, an Israeli army spokeswoman, said: "The air force just attacked over 40 tunnels found on the Gaza side of the border.
"We believe [they] were used for smuggling weapons, explosives and sometimes people," she said. "The pilots notified direct hits on these targets."
Gunfire was heard close to the Egyptian border with reports suggesting that Palestinians were attempting to break through, while the aerial bombardment continued over Gaza City.
Ayman Mohyeldin, Al Jazeera's correspondent in Rafah, a town split in half by the border, said at least one person died and 42 others were injured in the strikes on the tunnels.
"It's certainly a devastating blow to the civilian population in Gaza," he said, adding that speculation the tunnels might be hit had already caused the price of fuel and other goods to soar.
At the Rafah border, Palestinian fighters traded fire with Egyptian security forces, our correspondent said.
At least one Egyptian border guard and one Palestinian youth were killed in the clashes.
Tensions at the crossing with Egypt, bypassing Israel, had risen during the day, with Egypt blaming Hamas for not letting wounded Palestinians through and Hamas asking for medical aid to be handed over.
"We are ready for anything. If it's necessary to deploy ground forces to defend our citizens, we will do so"
Spokesman for Israeli defence minister
A Gaza health ministry official at the border, Alaa el-Din Mohammed el-Batta, said that transporting the seriously wounded was difficult and further complicated by Israeli air assaults.
"We have 25 in very critical condition," he said. "Because of the distance, there are fears that many will die on their way to Cairo.
A security official said that an Egyptian plane with 50 doctors on board as well as medical supplies had arrived in el-Arish near the border with Gaza.
Two Qatari aircraft carrying 50 tonnes of medical supplies have been waiting at the same airport.
Saudi Arabia's King Abdullah has ordered three plane-loads of medical aid sent to the Gaza Strip, the MENA news agency reported.
Iran says it is sending plane-loads of food to Cairo to be taken by the Egyptian Red Crescent to Gaza.
Haniya office hit
Earlier on Sunday afternoon, Israeli forces struck east of Gaza City, in Khan Yunis, and Jabaliya, in the north.
|Smoke billows from bombed tunnels in Rafah on Gaza's border with Egypt [AFP]
A police station and a factory were among the sites reportedly hit, after a mosque and the headquarters of al-Aqsa television were struck overnight.
The Reuters news agency said that at least one missile hit the offices of Ismail Haniya, the Hamas leader in the Gaza Strip, but he was not in the building at the time.
Ehud Barak, Israel's defence minister, warned that the air raids could be followed by a ground incursion.
"We are ready for anything. If it's necessary to deploy ground forces to defend our citizens, we will do so," Barak's spokesman quoted him as saying on Sunday.
Israeli television has reported that hundreds of infantry and armoured forces were massing on the border of the territory, and on Sunday the army was given approval to call up reservists to bolster its fighting strength.
Mustafa Barghouthi, a member of the Palestinian Legislative Council, rejected the Israeli government's claims that the air raids were in self-defence.
"This is a bloodbath, the bloodiest bloodbath since 1967," he told Al Jazeera. "This is an attack on the civilian population of Gaza."
Many of the dead in Saturday's attacks were police officers, including Tawfiq Jabber, the Gaza chief of police.
Ofir Gendelman, a spokesman for the Israeli foreign ministry, suggested that casualty figures put forward by the Palestinians were misleading and insisted that only Hamas targets had been hit.
"Hamas is using figures to attract public attention, media attention and for propaganda purposes," he told Al Jazeera.
"At the end of the day we are attacking Hamas strongholds ... No civilian targets are hit, it is very unfortunate that some civilians will be hit."
Hospitals, already suffering from shortages due to an 18-month blockade on the Gaza Strip, said they were struggling to cope with the number of injured, which includes women and children.
One of the buildings hit on Sunday was reportedly a warehouse used to supply local pharmacies with medicines.
A six-month truce between Israel and Hamas in the Gaza Strip ended on December 19.
Israel said it began its aerial assault on Gaza in response to rocket attacks launched by Hamas fighters into the south of the country.