TREASURIES-US yields climb after data; 10-yr set for monthly drop

* Job openings rose 9,000 to 7.594 million in May

* 10-yr yield set for first monthly drop in four

* Two-year yield set for fourth straight monthly climb (Updates to afternoon New York trading)

By Chuck Mikolajczak

NEW YORK, June 30 (Reuters) - U.S. Treasury yields advanced on Tuesday, although the benchmark 10-year note was poised to snap a three-month streak of gains after a reading on the labor market and as investors eyed developments in possible peace talks between the U.S. and Iran. Top U.S. envoys who have arrived in Doha will not hold a high-level meeting with Iran, a Qatari official said on Tuesday, casting doubt on the progress of efforts to bring a lasting halt to the Iran war and fully reopen the Strait of Hormuz. Yields briefly extended gains after the Labor Department said in its Job Openings and Labor Turnover Survey, or JOLTS report, that job openings, a measure of labor demand, had increased 9,000 to 7.594 million by the last day of May, above the 7.30 million estimate of economists polled by Reuters.

The data is the first in a string of reports on the labor market this week, culminating with Thursday's government payrolls report.

Yields have been declining in recent days until this week, 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.

"Today's number, and most critically, Thursday's number, it's really going to point a direction for the Fed," said Jack Ablin, chief investment strategist and founding partner at Cresset Capital Management in Chicago.

"I'm hopeful the jobs report on Thursday comes in kind of mildly positive, that would be best result. But if we get another strong result like this we could see Fed tightening expectations really ratchet up."

TREASURY YIELDS ADVANCE

The yield on the benchmark U.S. 10-year Treasury note climbed 4 basis points to 4.414%. Despite the monthly decline, the yield was set for its third-straight quarterly advance.

U.S. markets are closed on Friday for the July 4 Independence Day holiday.

Federal Reserve Bank of Cleveland President Beth Hammack said it remains possible that she'll advocate for higher interest rates if inflation pressures don't moderate, the latest in a series of comments from Fed officials recently that highlighted inflation.

The yield on the 30-year bond rose 4.1 basis points to 4.901% and was also on track to snap a three-month streak of gains and a third consecutive quarterly rise.

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 28.3 basis points.

The two-year U.S. Treasury yield, which typically moves in step with interest rate expectations for the Fed, ticked up 2 basis points to 4.129% and was on track for a fourth straight monthly gain, its longest since early 2022 and second straight quarterly gain. Markets are pricing in a 33.7% chance for a rate hike of at least 25 basis points at the Fed's July meeting, and a 66.9% at the September meeting, according to CME FedWatch.

The breakeven rate on five-year U.S. Treasury Inflation-Protected Securities was last at 2.27% after closing at 2.249% on Monday.

The 10-year TIPS breakeven rate was last at 2.236%, indicating the market sees inflation averaging about 2.2% a year for the next decade.

(Reporting by Chuck Mikolajczak; Editing by Nick Zieminski and Daniel Wallis)