Brent crude futures were flat at $89.42 per barrel at 0010 GMT, while U.S. West Texas Intermediate (WTI) crude futures rose 2 cents to $85.25.
Hamas said on Tuesday that an Israeli proposal on a ceasefire in their war in Gaza did not meet the demands of Palestinian militant factions, but it would study the offer further and deliver its response to mediators.
If the conflict continues, it risks the involvement of other countries in the region, particularly Hamas backer Iran, the third-largest producer in the Organization of the Petroleum Exporting Countries (OPEC).
Meanwhile, U.S. crude stocks climbed last week by 3.03 million barrels, according to market sources citing American Petroleum Institute figures. Analysts had estimated that stocks would rise by about 2.4 million barrels.
Official U.S. government inventory data is due at 10:30 a.m. (1430 GMT).
Separately, the government raised its forecast for U.S. crude oil output, expecting an increase of 280,000 bpd to 13.21 million bpd in 2024, up 20,000 bpd from an earlier forecast from the U.S. Energy Information Administration (EIA).
However, EIA said it expects Brent crude prices to average $88.55 a barrel in 2024, up from a previous forecast of $87 a barrel.
On Tuesday, both Brent and WTI fell more than 1%, as Israel-Hamas ceasefire discussions in Cairo continued.
The commander of the Revolutionary Guard's navy in Iran said it could close the Strait of Hormuz if deemed necessary. About a fifth of the volume of the world's total oil consumption passes through the strait daily.
Turkey said it would restrict exports of various products, including jet fuel, to Israel until there is a ceasefire. Israel said it would respond with its own curbs.
