* Markets pricing in roughly 30% chance of Fed rate hike

* 7-year note auction seen as average

* 10-year, 30-year yields down for third straight day

(Updates to afternoon trading)

By Chuck Mikolajczak

NEW YORK, July 28 (Reuters) - U.S. Treasury yields fell on

Tuesday, with the benchmark 10-year note on track for its first

three-day decline in a month, as oil prices continued to ease on

hopes for a lasting resolution to the U.S.-Iran conflict.

Traders were also bracing for the release of the Federal

Reserve's latest interest rate decision on Wednesday.

U.S. crude fell 3.68% to $79.55 a barrel and Brent

dropped to $84.58 per barrel, down 4.23% on the day

after falling more than 5% to a two-week low. Brent had topped

$100 a barrel last week as tensions in the Gulf increased, but

crude prices have since reversed on signs of cooling

hostilities.

U.S. President Donald Trump said in a Fox News interview on

Tuesday that there have been good talks with Iran but reiterated

threats to target a fortified underground facility near one of

Tehran's main nuclear sites as well as bridges and other

civilian targets if a deal is not made.

A Gulf source and a Western diplomat told Reuters that Oman

has presented Iran with a plan backed by Gulf states to manage

the Strait of Hormuz, including collecting voluntary fees for

using it.

BENCHMARK YIELDS HIT ONE-WEEK LOW

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

fell 3.9 basis points to 4.602% after hitting a

one-week low of 4.588%. The yield has fallen about 10 basis

points over the past three sessions.

"With lower oil, less forward inflationary pressure, so then

that helps rates back off a bit," said JoAnne Bianco, partner

and senior investment strategist at BondBloxx Investment

Management in Chicago.

"From the perspective of disruption in oil supplies or much

higher sustained oil prices, that's associated with the

resumption of the conflict," said Bianco, who noted a mitigation

of the conflict will cause oil prices to ease.

The yield on the 30-year bond shed 3 basis

points to 5.095% and was also on track for a third straight

drop, which would mark its longest run of declines since

mid-June.

Even with the recent drops, yields on 10-year and 30-year

bonds are poised for their biggest monthly increase since March,

the first full month of the Iran war, as the turn higher in oil

prices pushed up expectations for rate hikes from the Fed. The

2-year yield is on pace for a fifth straight monthly

rise, its longest since a nine-month climb that ended in April

2022.

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

measuring the gap between yields on 2- and 10-year Treasury

notes, seen as an indicator of economic

expectations, was at a positive 32.3 basis points.

MARKET EYES WEDNESDAY POLICY DECISION

Expectations that the Fed will raise its benchmark interest

rate by 25 basis points when it releases its latest policy

statement on Wednesday stand at 29.4%, according to CME Group's

FedWatch tool, up from 25.7% a week ago, but there is likely a

higher bar to a rate hike than the market is currently pricing

in.

Analysts at BofA Global Research said in a note on Tuesday

that they expect the Fed to hold rates steady with dissents from

two committee members but they do see "strategic incentives" for

Chairman Kevin Warsh to hike as it would differentiate him from

former Chair Jerome Powell and "allow him to claim credit for

any disinflation down the line, even if it's mechanical."

The 2-year U.S. Treasury yield, which typically

moves in step with interest rate expectations for the Fed,

dropped 4.8 basis points to 4.275% and was poised for its

biggest daily drop since July 15.

A $44 billion auction of 7-year notes was seen as mediocre,

with demand of 2.49 times the notes on sale roughly even with

the average, according to analysts.

The breakeven rate on 5-year U.S. Treasury

Inflation-Protected Securities (TIPS) was last at

2.169% after closing at 2.188% on Monday, its lowest close since

November 2024.

The 10-year TIPS breakeven rate was last at

2.195%, indicating the market sees inflation averaging about

2.2% a year for the next decade.