The military wing of the governing Palestinian party, Hamas, has said it will attack targets inside Israel if Israel does not end its Gaza offensive.
Hamas is threatening to resume attacks on targets inside Israel
However, Israeli Prime Minister Ehud Olmert has ordered his military to intensify its action there.
Hours later Israeli aircraft hit buildings in Gaza City, which the military said were used by militants.
Israel launched its assault on Tuesday night, to try to free a 19-year-old soldier captured a week ago.
Palestinian government buildings, bridges, a school, a university and a power plant have been hit in Israeli air strikes.
"I have given instructions to intensify the strength of action by the army and security services, to hunt down these terrorists, those who send them... and those who harbour them," Mr Olmert was quoted by Reuters news agency as telling his cabinet.
But if the soldier is released "the military activity that began with the kidnapping will stop", an unnamed Israeli official quoted Mr Olmert as saying, the Associated Press news agency reported.
On Sunday night, three Palestinian gunmen were shot dead by Israeli troops stationed near Dahaniya airport in south-east Gaza.
Earlier, a Hamas militant was killed in an Israeli missile strike in northern Gaza. The latest deaths bring to five the number of militants killed since Israel began its offensive.
Continuing its assault on Sunday, Israeli missiles struck the Gaza City office of Palestinian Prime Minister Ismail Haniya, setting it ablaze.
Palestinian officials said other targets included a school in Gaza City and Hamas facilities in northern Gaza.
In a statement, Hamas' military wing, Izzedine al-Qassam Brigades, threatened like-for-like attacks in Israel.
If [Israel] continue with these attacks [on Gaza] we will strike similar targets in the Zionist Occupation which we have not targeted until now," it said.
In the past, Hamas suicide bombers have struck many times in Israeli cities, the last time in Beersheba in August 2004, when 16 people died.
The last suicide bombing in Israel, which killed 11 people in Tel Aviv in April this year, was claimed by the smaller Islamic Jihad group.
The BBC's Alan Johnston in Gaza says the threat to resume attacks might turn out to be no more than rhetoric, but it will be taken as another sign of a hardening of the mood as the crisis surrounding the captured soldier escalates.
Cpl Gilad Shalit, a tank gunner, was captured in a cross-border attack claimed by three Palestinian militant groups last Sunday.
In an act of defiance, Mr Haniya held a cabinet meeting in the remains of his office, hours after it was damaged by two missiles.
The prime minister urged the "international community and the Arab League to take its responsibilities towards our people and intervene" to end what he called Israel's "insane policy", AFP news agency quoted him as saying.
GAZA CRISIS TIMELINE
Sun 25 June: Cpl Gilad Shalit captured in cross-border attack
Mon 26 June: Palestinian Popular Resistance Committees demand prisoner releases in exchange for Shalit
Weds 28 June: Israeli military enters southern strip after launching air strikes on Gaza
Thurs 29 June: Israel detains dozens of Hamas officials
Sat 1 July: Groups believed to be holding Cpl Shalit demand 1,000 prisoners be released
Israeli government spokesman Mark Regev said Mr Haniya's office was attacked as a warning to the Hamas leadership.
"It was a symbolic move because it was done at night and the office was vacant and no-one was hurt," he told the BBC.
"I think we wanted to send a message. [Ismail Haniya] is one of the leaders of Hamas, and we believe that he is responsible for what his organisation does and that when he says he knows nothing about this, I'm afraid we think it's being disingenuous."
On Sunday, Israel temporarily re-opened its main Karni cargo crossing into Gaza, allowing food, fuel and medical supplies to be sent in.
Israeli troops remain entrenched in south-east Gaza, where they entered on Wednesday in the first such offensive there since Israel pulled its troops and settlers out last September.
Israel has built up a large force on the border of the northern Gaza Strip but has yet to order its advance.