CAIRO, Aug 2 (Reuters) - Israeli airstrikes continued in Gaza for a second straight day on Sunday, killing at least four Palestinians, despite U.S. President Donald Trump's announcement of a breakthrough in efforts to implement last year's Gaza ceasefire agreement.

Palestinian health officials said Israeli warplanes carried out separate strikes in Gaza City and Deir al-Balah.

A man and his wife were killed and four people were wounded in an airstrike on an apartment in Deir al-Balah in the central Gaza Strip, while two more people, including a child, were killed in a strike on an apartment in Gaza City, medics said.

Trump said on Thursday there had been a breakthrough after Hamas agreed to disarm under a U.S.-backed initiative to implement the ceasefire agreement reached last year in Egypt. On Friday, Trump's Board of Peace published a 15-point roadmap setting out the final steps for implementing the agreement.

The roadmap has yet to be implemented. Hamas has said it will hand over its weapons for storage only after Israel halts military operations and withdraws its forces in line with last year's agreement. An Israeli official told Reuters there would be no withdrawal from the military's current positions unless Hamas undergoes "genuine disarmament."

Israeli Prime Minister Benjamin Netanyahu has not commented publicly on the initiative, while National Security Minister Itamar Ben-Gvir has called it unacceptable and said Israel should continue assassinating Hamas leaders.

Former senior Fatah official Mohammed Dahlan, who is based in the United Arab Emirates, said in a Facebook post that Jared Kushner, Trump's son-in-law and a senior adviser involved in the U.S. initiative, had told him he was working with the Israeli side to stop attacks on Gaza.

Dahlan said contacts with the U.S. were continuing to ensure the agreement was fully implemented, adding that its success now depended on Israel fully ending its daily attacks on Gaza.

Reuters has reached out to the U.S. State Department for comment on Dahlan's remarks. There was no immediate comment from Israel.

(Reporting by Nidal Al-Mughrabi and Hatem MaherEditing by Rod Nickel)