Pro-Hamas media said early on Sunday the airstrike killed Bardaweel, who is a member of the group's political office, and also killed his wife.
Israeli officials had no immediate comment.
Taher Al-Nono, the media advisor of the Hamas leadership, mourned Bardaweel's death in a post on his Facebook page.
After two months of relative calm, Gazans were again fleeing for their lives after Israel effectively abandoned a ceasefire, launching a new all-out air and ground campaign on Tuesday against Gaza's dominant Palestinian militant group Hamas.
Explosions echoed throughout the north, central, and southern Gaza Strip in the early hours of Sunday, as Israeli planes hit several targets in those areas in what witnesses said was an escalation of the attack that began on Tuesday.
In a statement, Hamas accused Israel of assassinating Bardaweel, whom it said was praying along with his wife, when an Israeli missile struck their tent shelter in Khan Younis.
"His blood, that of his wife and martyrs, will remain fuelling the battle of liberation and independence. The criminal enemy will not break our determination and will," said the group.
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.
The State of Palestine's mission to the United Nation's posted on X, saying Israel was intensifying its forced displacement campaign in the Gaza Strip.
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.
Palestinian medics said an Israeli plane bombed a house in the city of Rafah in the southern Gaza Strip, wounding several people.
Hamas has accused Israel of breaking the terms of the January ceasefire agreement by refusing to begin negotiations for a final end to the war and a withdrawal of its troops from Gaza.
But Hamas has said it is still willing to negotiate and was studying US President Donald Trump's special envoy Steve Witkoff's "bridging" proposals.
The return to the air strikes and ground operations that have devastated Gaza has drawn calls for a ceasefire from Arab and European countries. Britain, France and Germany issued a joint statement calling on Israel to restore access for humanitarian aid.
Israel has blocked the entry of goods into Gaza, and Netanyahu's foreign policy adviser Ophir Falk, accused Hamas of taking aid for its own use, a charge Hamas has previously denied.
Israel launched its campaign in Gaza after a devastating Hamas attack on Israeli communities around the Gaza Strip on October 7, 2023 that killed some 1200 people, according to an Israeli tally, and saw 251 abducted as hostages.
The Israeli campaign has killed more than 49,000 Palestinians, according to Palestinian health authorities, and devastated much of the coastal enclave, leaving hundreds of thousands of people in tents and makeshift shelters.
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