A force of Israeli tanks and armoured vehicles has moved into northern Gaza and is re-occupying former Israeli settlements in the area, reports say.
The Israeli operation would establish a border buffer zone
The push came after a second rocket hit the southern Israeli city of Ashkelon and follows an Israeli decision to step up attacks on the Hamas-led government.
A Hamas militant and a Palestinian policeman in northern Gaza were killed overnight, hospital sources said.
Israel has been raiding Gaza since the capture of a soldier on 25 June.
Before the first rocket attack on Tuesday, Ashkelon was thought to be out of range of militant rockets, which usually fall just inside the border with Israel.
Nobody was hurt in the attack, claimed by the military wing of Hamas, the Islamist organisation governing the Palestinian territories since January.
SOLDIER CAPTURE TIMELINE
Sun 25 June: Cpl Shalit captured in cross-border attack
Mon 26 June: Palestinian captors propose prisoner exchange
Weds 28 June: Israel enters southern Gaza after air strikes
Thurs 29 June: Israel detains dozens of Hamas officials
Sat 1 July: Captors demand 1,000 prisoners be released
Mon 3 July: Captors issue deadline of 0300 4 July
Tues 4 July: Israel: "Shalit still alive". Rocket hits Ashkelon
Wed 5 July: Second rocket hits Ashkelon. Israeli tanks enter northern Gaza
Confirming Israel's military response to the attacks, government minister Binyamin Ben-Eliezer said there was no wish to stay in Gaza, from where the Israeli military withdrew in 2005.
"Our presence there doesn't mean that we intend to remain in the Gaza Strip. We simply want to prevent firing at our towns," Mr Ben-Eliezer told Army Radio.
The military warned civilians to stay clear of the area until Israel's mission there was complete.
The BBC's Alan Johnston in Gaza said there is much speculation that Israel will aim to seize a swathe of the north of the territory, including three former settlements abandoned last summer.
Witnesses said troops had entered the abandoned Gaza settlements of Nissanit, Dugit and Elei Sinai.
Our correspondent said the aim would be to create a buffer zone and keep militants who fire rockets at Israeli towns further from the border.
Every night since the militants captured the soldier, the Israeli air force has struck at sites in Gaza, including bridges and a power station.
Little is known about the fate of Cpl Gilad Shalit, captured in a cross-border raid by militants on 25 June, but Israeli officials believe he is still alive.
In other developments on Wednesday, a Palestinian police officer and a member of an armed wing of Hamas who was laying a mine were killed by an Israeli shell in northern Gaza.
Also, a car carrying al-Jazeera reporters reportedly got caught in cross-fire between Israeli troops and gunmen near the Gaza town of Beit Lahiya, and their Palestinian driver was injured.
Israeli troops pulled out of the 45km-long territory, home to 1.4 million Palestinians, last summer after an occupation lasting 38 years.