Brent crude futures rose 16 cents, or 0.2%, to $82.69 a barrel by 0106 GMT, while U.S. West Texas Intermediate crude futures (WTI) climbed 15 cents, or 0.2%, to $77.73 a barrel.
Both benchmarks had settled more than 1% higher on Monday.
Iran-aligned Houthis have continued their attacks on shipping in the Red Sea, and while the Israel-Hamas war has not significantly constrained oil supply, it has increased freight rates and shipping time, leaving barrels on the water for longer.
U.S. President Joe Biden said on Monday he hopes to have a ceasefire in the Israel-Hamas conflict in Gaza start by next Monday as the warring parties appeared to close in on a deal during negotiations in Qatar that also aim to broker the release of hostages.
In public, Israel and Hamas continued to take positions far apart on a possible truce, while blaming each other for delays. Meanwhile, Kansas City Federal Reserve Bank President Jeffrey Schmid on Monday used a debut speech on policy to signal that he, like most of his central banking colleagues, is in no rush to cut interest rates. High borrowing costs typically reduce economic growth and oil demand. (Reporting by Arathy Somasekhar in Houston; Editing by Muralikumar Anantharaman)
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