June 4 (Reuters) - Benchmark U.S. Treasury yields fell

to an almost three-week low on Tuesday after data showed that

job openings fell more than expected in April, before highly

anticipated jobs data on Friday may give fresh clues on Federal

Reserve policy.

Job openings, a measure of labor demand, were down 296,000

to 8.059 million on the last day of April, the lowest level

since February 2021.

The report "shows softening, or increasing balance, in the

labor market, which is what the Fed wants to see and what the

market wants to see," said Vail Hartman, U.S. rates strategist

at BMO Capital Markets in New York.

Investors are closely watching economic releases as Fed

policy remains heavily data dependent. Traders see the U.S.

central bank as likely to begin cutting rates as soon as

September if inflation continues to moderate closer to its 2%

annual target and if economic growth cools.

"There's a lot of sensitivity to the data because of how

dependent the Fed is to the data," said Zachary Griffiths,

senior investment grade strategist at CreditSights in Charlotte,

North Carolina.

Treasury yields fell sharply on Monday after data showed

that U.S. manufacturing slowed for a second straight month in

May as new goods orders dropped by the most in nearly two years.

That may indicate that there is "more sensitivity to the

downside in yields," Griffiths noted, given "that's been an

indicator that's been very negative for almost two years now."

Benchmark 10-year note yields were last down 5

basis points at 4.353% and got as low as 4.324%, the lowest

since May 16.

Two-year note yields fell 4 basis points to

4.783% and reached 4.756%, also the lowest since May 16.

The inversion in the two-year, 10-year yield curve

deepened 1 basis points to minus 43 basis points.

Higher-than-expected inflation and strong growth in the

first quarter led investors to adjust for the likelihood that

the Fed will need to keep rates higher for longer to cool the

economy and bring down price pressures. Some softer data reports

in April, however, have boosted confidence that the economy may

slow enough for monetary easing this year.

This week's main U.S. economic release will be jobs data for

May due on Friday, which is expected to show that employers

added 185,000 jobs in May. It comes after April's report showed

that jobs growth slowed more than expected, with 175,000 jobs

gains, the fewest in six months.

"If that were to show more tightness in the labor market and

perhaps a pause in some of the progress toward better balance in

the labor market I could see this move reversing fairly

quickly," with yields heading higher, Griffiths said.

A softer number, however, could also extend the move lower

in yields, Griffiths added.

Other data this week will include the ADP National

Employment report and service sector data for May, both due on

Wednesday.

Next week's consumer price index (CPI) for May will then be

critical for guiding Fed expectations in the near-term. It will

come on Wednesday morning before the Fed is due to complete its

two-day policy meeting, when Fed officials will update their

economic and interest rate projections, which is known as the

"dot plot."

"If we're fairly consensus across headline payrolls, average

hourly earnings and unemployment rate, then I think that the

onus will be put on CPI next Wednesday in terms of solidifying

whether or not we're going to see 25 basis points or 50 basis

points signaled before year end in the dot plot," Hartman said.