* Crude prices climb, but gains capped by shipping optimism

* US labor market data due later this week

* Yield on 2-year note rises after four straight daily

declines

(Updates prices and commentary)

By Chuck Mikolajczak

NEW YORK, June 29 (Reuters) - U.S. Treasury yields edged

higher on Monday, as crude prices rose following attacks between

the U.S. and Iran over the weekend and ahead of a flurry of

labor market data later this week.

U.S. crude rose 2.02% to $70.62 a barrel and Brent

rose to $72.92 per barrel, up 1.29% on the day as the

attacks once again threatened a tenuous peace deal, although

expectations of a continued recovery in energy shipping through

the Strait of Hormuz kept gains in check.

Iranian and U.S. technical teams working on the implementation

of an interim peace deal are expected to meet in Doha in the

coming days, a source told Reuters on Monday.

Markets will see a string of data on the labor market this

week, culminating with the release on Thursday of the Labor

Department's monthly payrolls report for June.

Markets are closed on Friday for the U.S. Independence Day

holiday on July 4.

Yields have been declining in recent days as expectations of

easing inflation pressures have grown with a decline in oil

prices, offsetting what was seen as a hawkish Federal Reserve

policy announcement and press conference by new Fed Chairman

Kevin Warsh on June 17.

"The labor market to me is really interesting, the data for

this week, but the key, the focus now is more on the inflation

side," said Jim Barnes, director of fixed income at Bryn Mawr

Trust.

"Because energy prices have materially come down, inflation

expectations have notably come down. But the market now,

especially after the Fed meeting, it's not good enough, now they

have to start to see more concrete evidence that inflation's

coming down."

BENCHMARK YIELDS EDGE HIGHER

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

edged up 0.6 basis point to 4.378% after having

fallen for three straight weeks.

Comments from several Fed officials indicated last week that

they were still concerned about high inflation.

The yield on the 30-year bond shed 0.3 basis

point to 4.862%.

Barclays analysts on Monday said in a note that AI-related

categories have been putting upward pressure on inflation as

measured by the personal consumption expenditures price index

(PCE) and while price gains may moderate as supply increases,

"the Fed will likely find it hard to dismiss AI-related

inflation entirely."

Markets are currently pricing in a 31.5% chance of a rate hike

of at least 25 basis points at the Fed's July 28-29 meeting,

according to CME Group's FedWatch tool, and a 62.9% chance at

the September 15-16 meeting.

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

typically moves in step with interest rate expectations for the

Fed, rose 2.5 basis points to 4.113% and was on track for its

first daily gain after four straight declines.

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 26.1 basis points.

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

Inflation-Protected Securities (TIPS) was last at

2.256% after closing at 2.223% on Friday.

The 10-year TIPS breakeven rate was last at

2.224%, indicating the market sees inflation averaging about

2.2% a year for the next decade.