Despite reports that Gaza and Israel had reached an Egypt-brokered ceasefire after rocket strikes injured at least 13 people on Monday, both sides appeared to have breached the truce.
Sky News Arabia reported that since the ceasefire was announced, there have been two breaches by Israel.
Militant factions in Gaza told Turkish news agency Anadolu that they would adhere to the ceasefire if Israel does.
Haaretz also reported that since the ceasefire, Israel's Iron Dome air defence system has intercepted rockets over communities close to the Gaza border, after sirens sounded in the Sderot, Eshkol and Sha'ar HaNegev neighbourhoods. Palestinian factions said that the latest barrage of rockets was in response to Israeli attacks.
The UN Security Council announced a meeting for Tuesday to discuss the Gaza situation.
The Israeli military said that it had begun carrying out strikes on Hamas targets in the Gaza Strip, hours after a Palestinian rocket hit a house near Tel Aviv.
A Hamas naval post west of Gaza City and a large training camp in northern Gaza were hit, Palestinian security officials and Hamas media outlets said.
Because Hamas had hours of notice that Israeli strikes were coming, both posts were probably evacuated.
The announcement followed the worst round of fighting between Hamas and Israel since the Gaza war in 2014.
Israel reportedly struck Hamas security headquarters and the office of its leader, Ismail Haniyeh, inside Gaza.
Six Palestinians were injured in the air strikes, Sky News Arabia reported.
The attacks were in retaliation to a rocket fired from Gaza early on Monday north of Tel Aviv, striking a house and injuring seven Israelis.
The strike on Israel caused Prime Minister Benjamin Netanyahu to cut short his visit to the US and command his military to respond forcefully on Hamas targets, sparking fears of a new escalation.
The trading of fire comes 10 days after rockets were launched towards the commercial capital of Israel, and at a time of high tension before the Gaza border protest anniversary on Saturday and the Israeli election on April 9.
Sirens sounded in central Israel on Monday and medics responded to a burning house near Tel Aviv, where they said six people were wounded after a suspected rocket attack from the Gaza Strip.
The Israeli army accused Hamas and other militant factions in Gaza of firing barrages of rockets into Israel on Monday night. Some were intercepted by the Iron Dome.
Hamas denied that it was responsible for the rocket attack but that did not appear to matter on Monday night.
A Hamas official denied to AFP that the group fired a rocket, saying it could have been "bad weather".
In Lebanon, Hezbollah Secretary General Hassan Nasrallah met a delegation from Hamas led by Saleh Al Arouri.
In recent years, splinter groups have fired rockets into Israel, but the country blames Hamas for any fire from the strip.
"As we speak, Israel is responding forcefully to this wanton aggression," Mr Netanyahu said, standing next to US President Donald Trump in Washington.
Mr Trump had just formally recognised the occupied Golan Heights as Israeli territory, breaking with decades of US policy.
Hamas's leader said his group would respond if Israel retaliated too forcefully to the overnight rocket strike.
Mr Haniyeh said that the Palestinian people "will not surrender" and its militant factions "will deter the enemy if it exceeds the red lines".
His statement came as Israel began striking targets in the Gaza Strip.
Hamas's leaders went into hiding, police and security installations were evacuated and hospitals in Gaza were on alert in anticipation of the air strikes.
Mr Netanyahu was in Washington to meet Mr Trump in the home stretch of a tight Israeli parliamentary election campaign.
"In light of the security events I decided to cut short my visit to the US," he said, calling the attack a heinous crime.
The early morning rocket siren was heard in Mishmeret, an agricultural town north of Tel Aviv in the Emek Hefer region.
The Magen David Adom ambulance service said it treated seven of the house's occupants for wounds.
If confirmed, it would be one of the farthest rocket launches from Gaza against Israel since even before the 2014 conflict.
In that seven-week war, the longest distance reached by Gazan rockets was to Tel Aviv, 20 kilometres closer to the enclave than Mishmeret.
On Sunday, Mr Netanyahu said he could go to war with Gazan groups in the election campaign period if he had to.
"Is it worth it now to enter Gaza? I am not sure. We are experiencing our best decade security wise," he told Israel Radio.
The remarks followed two days of Israeli air strikes on Hamas posts in the Gaza Strip.
Gaza is controlled by Hamas, which has fought three conflicts with Israel since 2008. The territory is home to other groups, including Islamic Jihad, an Iranian-backed armed organisation that has a formidable rocket arsenal.