Al Qassam Brigades of the Palestinian group Hamas had warned the residents of the Israeli city of Ashkelon to leave their homes by 5 pm local time, in response to Israel's displacement of Palestinians.
Hamas's statement on Telegram said, "In response to the crime of the enemy displacing our people and forcing them to flee their homes in several areas of the Gaza Strip, we give the residents of the occupied city of Ashkelon time to leave before five o'clock in the evening."
Hamas fighters claimed to have hit Tel Aviv's Ben Gurion airport as part of ongoing rocket attacks on Israel, however, the Israelis have refuted this.
Israel has responded by claiming that two top Hamas figures—its minister of economy, Jawad Abu Shamala, and Zakaria Abu Maamar, a member of Hamas's political office—were killed by its air force.
It comes after Israeli authorities said that 1,500 jihadi remains had been discovered on their soil during Saturday's unexpected strike.
More than 150 Israeli soldiers and civilians, according to Israel, have been abducted and are being held captive in Gaza.
Families have reportedly been abducted, leaving relatives in the dark about what happened to them.
Hamas has threatened to kill a hostage whenever Israel fires an airstrike in retaliation without giving civilians the time to flee.
According to Palestinian health officials, 4,000 people have been hurt in strikes so far, and 770 Palestinians have died in the Gaza Strip.
Hamas sources in Lebanon have told Sky News the "operation" on Saturday was in the planning for more than a year.
The source also went onto say that if Israeli Prime Minister Benjamin Netanyahu expands the front, Hamas will too and their allies "will not leave Hamas in this battle alone".