By Erwin Seba

HOUSTON (Reuters) -Oil prices slid on Tuesday, settling down more than 4% on news of a possible ceasefire between Hezbollah and Israel, although prices found some support on fears of a potential attack on Iranian oil infrastructure.

Brent crude futures settled down $3.75, or 4.63%, at $77.18 a barrel. U.S. West Texas Intermediate futures finished down $3.57, or 4.63%, at $73.57 a barrel. At their session lows, both were down more than $4 a barrel.

"We continue to be very headline dependent," said John Kilduff, partner with Again Capital LLC. "This morning, we heard about the potential ceasefire. Then we got indications targets are still being dialed in and energy targets are in the mix."

"That Hezbollah is open to a ceasefire, is the kind of headline that people jump on," said Phil Flynn, senior analyst at Price Futures Group. "There should be a lot of volatility up and down on this conflict."

On Monday, Brent rose above $80 per barrel for the first time since August after more than a 3% daily gain. That followed the largest weekly gain in over a year, roughly 8%, in the week to Friday on rising concerns of a spreading war in the Middle East.

Hezbollah left the door open to a negotiated ceasefire after Israeli forces raised the stakes in the conflict with its Iran-backed enemy by making new incursions in the south of Lebanon.

Israeli defense minister Yoav Gallant said it appeared the replacement for slain Hezbollah leader Sayyed Hassan Nasrallah had also been eliminated.

Late on Tuesday, Israel's military warned people away from specific buildings in the southern suburbs of Beirut.

The oil price rally began after Iran launched a missile barrage at Israel on Oct. 1. Israel has sworn to retaliate and said it was weighing its options.

Some analysts said an attack on Iranian oil infrastructure was unlikely and warned oil prices could face considerable downward pressure if Israel focuses on any other target.

In the U.S., Hurricane Milton intensified into a Category 5 storm on its way to Florida after forcing at least one oil and gas platform in the Gulf of Mexico to shut on Monday.

U.S. crude oil stocks rose by nearly 11 million barrels last week, while fuel stockpiles fell, according to market sources citing American Petroleum Institute figures on Tuesday.

Crude stocks rose by 10.96 million barrels in the week ended Oct. 4, the sources said on condition of anonymity. Gasoline inventories fell by 557,000 barrels, and distillate stocks fell by 2.60 million barrels, they said.

(Reporting by Erwin Seba in Houston; Additional reporting by Shariq Khan in New York, Arunima Kumar and Robert Harvey in London and Emily Chow in Singapore; Editing by David Evans, Mark Potter, Emelia Sithole-Matarise, Bill Berkrot and David Gregorio)