* June producer prices fell 0.3% after a downwardly revised

0.6% increase in May

* Markets pricing in 12.3% chance of July rate hike

* Fed's Williams says inflation may have crested

By Chuck Mikolajczak

NEW YORK, July 15 (Reuters) - U.S. Treasury yields fell on

Wednesday, with the benchmark 10-year Treasury note on track for

its first consecutive daily declines in nearly three weeks,

after economic data showed an easing of price pressures for a

second straight day.

The Labor Department said the Producer Price Index for final

demand dropped 0.3% last month, below the estimate of economists

polled by Reuters that called for an unchanged reading, after a

downwardly revised 0.6% increase in May.

In the 12 months through June, the PPI increased 5.5% after

rising 6.0% in May.

The softer-than-expected data followed the release on Tuesday of

the Consumer Price Index report, which showed inflation

moderated in June.

"What's going on is that the numbers look decent from the

standpoint of the Fed and what (Chairman Kevin) Warsh may do,

and for the most part, people are sort of not believing it

because they don't see any end to this conflict with Iran," said

Tom di Galoma, managing director of global rates trading at

Mischler Financial Group in Stamford, Connecticut.

"So the conflict is going to get worse and so the fall in

CPI and PPI is really kind of a temporary thing."

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

fell 2 basis points to 4.565% and was on pace for

its first back-to-back daily decline since June 26.

IRAN TENSIONS CLOUD OUTLOOK

Energy prices have come down in recent weeks on expectations

that a durable peace deal could be reached between the U.S. and

Iran. However, hostilities have intensified in recent days and

caused a reversal in crude prices.

U.S. crude advanced 0.68% to $79.88 a barrel and Brent

rose to $85.01 per barrel, up 0.33% on the day.

The recent inflation readings have led to a drop in

expectations that the Federal Reserve would raise interest rates

at its policy meeting later this month, with markets now pricing

in a 12.3% chance for a hike of at least 25 basis points, down

from more than 40% on Monday, according to CME Group's FedWatch

tool. Expectations for a hike at the September meeting, however,

are still roughly 50%.

The yield on the 30-year bond shed 0.2 basis

points to 5.092%.

CENTRAL BANK AWAITS MORE PROOF

Top Fed officials, including Warsh, on Tuesday welcomed the

cooler CPI data, but said they would need more such readings to

feel confident that price pressures are truly easing.

Warsh is testifying before the Senate Banking Committee on

Wednesday.

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 40.3 basis points.

On Wednesday, New York Fed President John Williams said that

while inflation is "unquestionably too high," there are reasons

to believe it may have crested and should soon start subsiding,

with monetary policy well positioned to guide inflation back to

the central bank's 2% target.

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

typically moves in step with interest rate expectations for the

Fed, fell 3.3 basis points to 4.16%.

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

Inflation-Protected Securities (TIPS) was last at

2.257% after closing at 2.284% on July 14.

The 10-year TIPS breakeven rate was last at

2.247%, indicating the market sees inflation averaging about

2.2% a year for the next decade.