Hamas said the air strike on Khan Younis killed Bardaweel and his wife.
The Israeli military confirmed in a statement on Sunday that it had killed the Hamas official on Saturday.
"This elimination further degrades Hamas' military and government capabilities," the military added.
After two months of relative calm in the war, Gazans have again been fleeing for their lives after Israel effectively abandoned a ceasefire, launching a new all-out air and ground campaign on Tuesday against Hamas.
Explosions echoed throughout the north, central and southern Gaza Strip early on Sunday, as Israeli planes hit several targets in those areas in what witnesses said was an escalation of the attacks that began earlier in the week.
At least 30 Palestinians had been killed in Israeli strikes on Rafah and Khan Younis so far on Sunday, health authorities said. Those killed included three municipal employees, medics said.
Bardaweel was a member of the Hamas decision-making body, the political office, and had held posts such as heading the Hamas delegation for indirect truce talks with Israel in 2009 and led the group's media office in 2005.
"His blood, that of his wife and martyrs, will remain fuelling the battle of liberation and independence," the group said.
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. Israel launched its initial assault on Gaza after Hamas fighters stormed southern Israel on October 7, 2023, killing 1200 people and taking more than 250 hostages, according to Israeli tallies.
Hamas has accused Israel of breaking the terms of the January ceasefire agreement by refusing to begin negotiations for an 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 "bridging" proposals from US President Donald Trump's special envoy, Steve Witkoff.
At least 50,021 Palestinians have been killed and 113,274 wounded since the beginning of the war, the health ministry said in a statement on Sunday.
Israeli military spokesperson Avichay Adraee issued an evacuation warning on X for residents in the Tel Al-Sultan neighbourhood in western Rafah in the south of the strip.
The military later said troops had encircled Tel Al-Sultan to dismantle "terror infrastructure sites and eliminate terrorists in the area", in order to reinforce control and expand the security zone in southern Gaza.
It said soldiers were allowing the evacuation of civilians from the combat zone via organised routes for their safety.
Dozens of families quit their homes in Tel Al-Sultan heading northward to Khan Younis, some on foot, while others carried their belongings and children on donkey carts and rickshaws.
"When the ceasefire began, we returned to put up tents next to the ruins of our homes, dreaming that soon our homes would be rebuilt," said Abu Khaled, a Rafah resident.
"Now we are fleeing under fire for maybe the 10th time, when will we ever rest? When will there ever be peace in this city?" he told Reuters via a chat app.
The Palestinian Civil Emergency Service said 50,000 residents remained trapped in Rafah after they were surprised by an Israeli army raid into their areas, warning their lives, and those of rescue teams, were at risk.
Palestinian and international officials also warned about the return of the risk of famine in the enclave.
"Every day without food inches Gaza closer to an acute hunger crisis. Banning aid is a collective punishment on Gaza: the vast majority of its population are children, women & ordinary men," the head of the United Nations agency on Palestinian refugees, UNRWA, Philippe Lazzarin posted on X.
On March 2, Israel blocked the entry of goods into Gaza, accusing Hamas of taking aid for its own use, a charge Hamas has previously denied.
