The US at first requested voting to take place on Tuesday afternoon instead of the morning because its mission to the United Nations wanted more time to consult the text of the resolution drafted by the United Arab Emirates (UAE), the source said on Tuesday.
However, Washington later said it wanted to delay a vote on the resolution until Wednesday morning, the source said.
The United States is the only member of the UN Security Council who has vetoed all previous resolutions on Gaza.
US diplomats have claimed Washington wants to see text in the resolutions condemning the Palestinian movement Hamas. Other UN Security Council members have opposed including such provisions in the resolutions.