BENGALURU/LONDON June 23 (Reuters) - Oil prices fell on Tuesday, extending losses from the previous session, on signs of some progress in restoring crude flows through the Strait of Hormuz following U.S.-Iran peace talks.

Brent crude futures were down 43 cents, or around 0.6%, to $77.47 a barrel and U.S. West Texas Intermediate was down 32 cents, or 0.4%, to $73.54 a barrel at 1001 GMT.

Prices fell more than 3% on Monday after the United States granted Iran a 60-day sanctions waiver following initial peace talks, and as officials reported a lull in hostilities in Lebanon under a broader agreement.

Venezuelan, Russian and now Iranian crude is available to anyone looking to buy, said Ole Hvalbye, market analyst at SEB Research, highlighting that countries would be looking to stock up on crude to replenish stores.

In the short term, the easing of sanctions wouldn't weigh on prices much, he added, as the U.S.-Iranian memorandum of understanding was still new and fragile. 

On Monday, government data showed U.S. crude stocks in the Strategic Petroleum Reserve fell to 331.2 million barrels last week, the lowest since June 1983.

Oil and liquefied natural gas tankers sailed through the Strait of Hormuz on Monday, in a sign of traffic slowly picking up even though Iran said it had again closed the waterway over the weekend, shipping data showed.

The world has lost millions of barrels of oil and gas supply since the war closed the strait, a chokepoint for about a fifth of the world's oil and LNG supplies, for more than three months.

"Ship owners and operators will require assurances that the threats posed by mines have been fully eliminated. Damaged ports, debris in the water, and congestion present additional obstacles to an unconditional ramp-up in traffic," said Tamas Varga, an analyst at PVM Oil Associates.

The price declines come after a weekend that had appeared to put the week-old accord in jeopardy, including threats from U.S. President Donald Trump to restart the war if Iran disrupted shipping again.

Rabobank cut its oil price forecasts, citing eased disruption risks in the Gulf, and now sees Brent at $79 a barrel in the third quarter and $78 in the fourth quarter.

Iraq further increased output from its southern oilfields to around 2.1 million barrels per day as more tankers line up to load crude from its Gulf export terminals, two Iraqi oil officials told Reuters.

Separately, analysts in a Reuters poll expect U.S. crude inventories to have fallen last week, along with distillate and gasoline inventories.