Israel has fought three wars with Palestinian militants in Gaza since 2008 and says the blockade is necessary to keep them from obtaining weapons or materials that could be used for military purposes.
"I am deeply alarmed by the recent escalation of violence between Gaza and Israel, and particularly by today's multiple rockets fired toward communities in southern Israel", said U.N. Middle East envoy Nickolay Mladenov in an overnight statement.
Israel launched strikes on Wednesday after dozens of rockets were fired into Israeli territory by Hamas.
Those killed in the Gaza Strip included Enas Khammash, 23, and her 18-month-old daughter Bayan, according to the health ministry in the Hamas-controlled enclave.
Injuries were also reported on both sides.
Meanwhile, an official from the Islamist Gaza Strip government told Turkey's state-run Anadolu Agency on Wednesday that Hamas may be ready to reach an agreement toward a truce with Israel.
Smoke plumes could be seen rising from Gaza City following the military's announcement it was targeting "terror sites in the Gaza Strip".
In a written statement, the Ezzeddin Qassam Brigades, the military wing of Hamas, claimed responsibility for the attacks.
Earlier on Wednesday, before the rocket fire, a Hamas official said these negotiations were in "advanced stages". The Iron Dome aerial defense system intercepted at least 21 launches that sent rockets heading for densely populated areas in the south; nevertheless, the projectiles that managed to find their targets did considerable damage.
Security forces belonging to Hamas inspect the site after Israeli army conducted air strikes towards Gaza City
There were no immediate reports of fresh Israeli injuries or damage.
He called on all sides to step "back from the brink".
United Nations and Egyptian-led efforts are under way to mediate a long-term truce between Israel and Hamas that would end a four-month surge of violence along the Israel-Gaza border.
The rocket fire came after two fighters from the military wing of Hamas were killed by Israeli fire on Tuesday.
The escalation comes after the Hamas leadership convened for a rare meeting in Gaza on Friday.
They also demand an end to Israel's 11-year blockade of the Gaza Strip, which has gutted the coastal enclave's economy and deprived its roughly two million inhabitants of many basic commodities.
The exchange came amid continued attempts by Egypt to broker a broad cease-fire deal between Israel and Hamas.
"The IDF sees with severity the terrorist activities of Hamas", said the army spokesperson in a statement Wednesday night.
At least 158 Palestinians have been killed in four months of weekly border protests and one Israeli soldier was shot dead by a Gaza sniper.