(Updates to afternoon New York trading)

* Jobless claims rise above forecasts

* Oil prices fall on hopes for US-Iran deal

* Fed officials signal inflation as priority

By Chuck Mikolajczak

NEW YORK, June 4 (Reuters) - U.S. Treasury yields fell

on Thursday after labor market data was softer than expected,

while oil prices retreated on renewed hopes that a deal to end

the U.S.-Israeli war with Iran could be reached.

Israel and Lebanon agreed to implement a ceasefire to end

hostilities, the Trump administration said on Wednesday, raising

hopes for a broader agreement to end the Iran war.

However, the pro-Iran Hezbollah movement rejected a new

ceasefire in Lebanon on Thursday and Israel said it would not

withdraw troops from the country, complicating peace efforts.

U.S. crude fell 3.1% to settle at $93.04 a barrel and

Brent dropped to $95.03 per barrel, to settle down

2.84%.

On the economic front, the Labor Department said weekly initial

jobless claims rose 13,000 to a seasonally adjusted 225,000,

topping the 213,000 estimate of economists polled by Reuters,

although the underlying trend indicated the labor market was on

stable footing.

"If we get a deal done, then we'll have a short-run

reprieve," said Thomas Urano, co-chief investment officer at

Sage Advisory in Austin.

"The background is we know we've got inflation pressure

that's still there and pretty constant ... and the only other

ballast that we've had so far has been the idea of a weak labor

market."

The claims data was the latest in a string of reports on the

labor market this week, with the key government payrolls report

scheduled for Friday.

TEN-YEAR TREASURY YIELD FALLS

The yield on the benchmark U.S. 10-year Treasury note

slipped 2 basis points to 4.471%.

Other data from the Labor Department showed worker productivity

increased at a downwardly revised 0.3% annualized rate last

quarter, the slowest pace since the first quarter of 2025 and

shy of the 0.5% estimate.

Unit labor costs increased at a 1.8% rate last quarter, a

downward revision from the 2.3% pace reported last month and

below the 2.5% forecast.

The yield on the 30-year bond declined 1.5

basis points to 4.975%.

A closely watched part of the U.S. Treasury yield curve

measuring the gap between yields on two- and 10-year Treasury

notes, seen as an indicator of economic

expectations, was at a positive 42.4 basis points.

The two-year U.S. Treasury yield, which

typically moves in step with interest rate expectations for the

Federal Reserve, lost 3.9 basis points to 4.045%.

San Francisco Federal Reserve President Mary Daly said that the

U.S. interest-rate path will depend on how the economy evolves,

adding that monetary policy is "in a good place" and the Fed is

prepared to respond "either way."

Kansas City Federal Reserve President Jeffrey Schmid said the

U.S. central bank's choice now is between being patient and

holding interest rates steady or hiking rates to tamp down

inflation that has been above the central bank's 2% target for

years.

After beginning the year pricing in about 50 basis points of

cuts from the Fed this year, market expectations have shifted

and are now pricing in nearly 20 basis points in hikes,

according to LSEG data.

The breakeven rate on five-year U.S. Treasury

Inflation-Protected Securities was last at 2.496%

after closing at 2.527% on Wednesday, its lowest close in three

months.

The 10-year TIPS breakeven rate was last at

2.378%, indicating the market sees inflation averaging about

2.4% a year for the next decade.