* US-Iran peace deal hopes drive oil prices lower

* Lower oil prices support Treasury yield decline

* Fed officials say inflation risks may delay rate cuts

* Initial jobless claims rise, but below expectations

By Chuck Mikolajczak

NEW YORK, May 7 (Reuters) - U.S. Treasury yields fell on

Thursday, with the benchmark 10-year note on track for a third

straight decline, as it tracked the recent drop in oil prices on

optimism that progress was being made on a peace deal in the

Iran war.

The United States and Iran are edging toward a limited,

temporary agreement to halt their war, sources and officials

said on Thursday, with a draft framework that would stop the

fighting but leave the most contentious issues unresolved.

U.S. crude fell 4.44% to $90.86 a barrel and Brent

fell to $96.91 per barrel, down 4.31% on the day,

although prices remained volatile.

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

shed 1.8 basis points to 4.336% and was on pace

for a third straight daily decline, which would mark its longest

streak in nearly three months.

"The bond market's been remarkably quiet throughout this

process, and in fact has not sold off as much as you would have

expected, given the reduction in expected cuts, so therefore,

it's just (going to) grind lower rather than just rip lower if

we get a resolution," said Jay Hatfield, chief executive and

chief investment officer at Infrastructure Capital Advisors in

New York.

"It just makes sense that we're on hold waiting for the

resolution of this proposal to Iran because it's really kind of

98% of the catalyst for the market over the next month,"

Hatfield said.

The yield on the 30-year bond slipped 0.7

basis point to 4.936%. The 30-year had hit 5.036% on Monday, its

highest since July 17.

Since the war began on February 28, yields have steadily

climbed as inflation worries dented market expectations for rate

cuts from the Federal Reserve this year.

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 48.5 basis points.

JOBLESS CLAIMS EDGE HIGHER

On the economic front, weekly initial jobless claims rose by

10,000 to a seasonally adjusted 200,000, slightly below the

205,000 estimate of economists polled by Reuters. The data was

the latest read on the labor market this week and comes ahead of

Friday's key government payrolls report.

Other Labor Department data showed nonfarm productivity

increased at a 0.8% annualized rate last quarter, below the 1.0%

estimate and down from the downwardly revised 1.6% rate in the

fourth quarter.

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

typically moves in step with interest rate expectations for the

Fed, fell 2.3 basis points to 3.849% after sliding to 3.824%,

its lowest since April 28.

Federal Reserve Bank of Cleveland President Beth Hammack

reiterated that risks of higher inflation in the future argued

against the U.S. central bank holding on to its policy leaning

toward cutting rates at some point.

Other Fed officials scheduled to speak on Thursday included

Bank of New York President John Williams and Bank of Minneapolis

President Neel Kashkari.

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

Inflation-Protected Securities (TIPS) was last at

2.571% after closing at 2.591% on May 6.

The 10-year TIPS breakeven rate was last at

2.42%, indicating the market sees inflation averaging about 2.4%

a year for the next decade.