Israel has faced international criticism for largely excluding Palestinians in the occupied West Bank and Gaza from its highly successful vaccination campaign.
The dispute highlights the Palestinians' reliance on Israel even as they struggle to combat the pandemic on their own.
Israel and Hamas have fought three wars and numerous skirmishes since the militant group took control of Gaza.
But Ashraf al-Qedra, spokesman for the Gaza Health Ministry, said they would be administered in two phases, first to vulnerable patients and then to health care workers.
Israel captured east Jerusalem, the West Bank and Gaza in the 1967 war, territories the Palestinians want for their future state.