Pro-Hamas media reported his death in the early hours of Sunday, saying the airstrike killed Bardaweel, who is a member of the group's political office, and also killed his wife.
Israel resumed significant strikes on Gaza on Tuesday, blaming Hamas, abandoning a ceasefire agreement that began on January 19 and ending almost two months of calm.
Israeli Prime Minister Benjamin Netanyahu has repeatedly said the main aim of the war is to destroy Hamas as a military and governing entity. He has said the aim of the new campaign is to force the group to give up remaining hostages.
Hamas' de facto government head Essam Addalees and internal security chief Mahmoud Abu Watfa were among those killed by Israeli strikes on Tuesday, in addition to several other officials.
Palestinian health officials said at least 400 people, more than half of them women and children, were killed on Tuesday.
The report comes after Israeli artillery and airstrikes hit south Lebanon on Saturday after Israel said it had intercepted rockets fired from across the border, killing at least eight people and endangering a shaky truce that ended a year-long war between Israel and Lebanese armed group Hezbollah.
That conflict marked the deadliest spillover of the Gaza war, and a blistering Israeli offensive after months of cross-border exchanges of fire wiped out Hezbollah's top commanders, many of its fighters and much of its arsenal.
Hezbollah denied responsibility for Saturday's strikes, saying it had "no link" to the rocket launches and that it remained committed to the ceasefire. No group claimed responsibility for the attack.
An Israeli official said the identity of the group which fired the rockets had not been confirmed. Six rockets were fired, the official said, three of which crossed into Israel and were intercepted.
Two waves of Israeli strikes killed three people in Bint Jbeil and Touline, as well as five in the port city of Tyre, all in southern Lebanon, according to Lebanon's state news agency, which cited health authorities.
Saturday's exchanges were the first since Israel in effect abandoned a separate ceasefire in Gaza with Palestinian militant group Hamas, an ally of Hezbollah, both backed by Israel's arch-foe Iran.
Later in the day, the Israeli military announced a second round of strikes on what it said were Hezbollah targets.
"We expect Lebanon to take care of its part of the agreement," Ophir Falk, Netanyahu's foreign policy adviser, told Reuters.
"The IDF will do whatever it takes to enforce the ceasefire and to make sure that our civilians can get back home safely and securely," Falk said.
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