June 6 (Reuters) - Benchmark U.S. 10-year Treasury yields edged up from two-month lows on Thursday as investors waited to see if Friday’s employment report for May would show signs of strength in the U.S. economy.

Bonds have rallied this week as investors have warmed to the view that the Federal Reserve could begin cutting rates as soon as September as the economy softens.

Inflation easing closer to the U.S. central bank’s 2% annual target is key to when the Fed cuts and how many rate reductions are likely this year.

“The next month or two in terms of data is going to be very, very telling,” said Scott McIntyre, senior portfolio manager at HilltopSecurities Asset Management in Austin, Texas. “We're at this pivot point where investors are wondering whether the hot first quarter numbers are going to cool in the second quarter,” McIntyre said. “We're just now getting numbers for May so that hasn't been determined yet.”

Benchmark 10-year note yields were last up 1 basis point on the day at 4.299%. They got as low as 4.275% on Wednesday, the lowest since April 1.

Two-year note yields were little changed on the day at 4.732% after earlier touching 4.720%, their lowest since May 16.

The inversion in the two-year, 10-year yield curve narrowed 1 basis point to minus 44 basis points.

Traders said that the absence of fresh Treasury supply had supported the market this week after some soft bond auctions last week.

However, investors have also positioned for softer jobs data on Friday, with the possibility that jobs gains will come in below the median economist forecast of 185,000 jobs.

April's report showed that jobs growth slowed more than expected, with 175,000 jobs gains, the fewest in six months.

Data this week has also pointed to more balance in the labor market.

The ADP Employment Report on Wednesday showed that private payrolls increased by 152,000 jobs last month, below economists’ forecasts for 175,000 in jobs gains.

A survey on Tuesday also showed that 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.

Data on Thursday showed that the number of Americans filing new claims for unemployment benefits increased last week. U.S. worker productivity also grew slightly less than previously estimated in the first quarter but exceeded market expectations, and unit labor costs rose by less than first thought.

Wage data in Friday’s jobs report will be closely watched as inflation remains the key focus for Fed policy.

“Average hourly earnings realistically are probably the most important component, I think, because that feeds into inflation, and inflation right now is priority one for the Fed,” said McIntyre.

Next week’s consumer price index (CPI) report for May will then be key to guiding near-term Fed expectations. Fed officials have stressed that they want to see several months of improving inflation before easing policy.

The consumer price inflation report will come on Wednesday, before the U.S. central bank is due to complete a two-day policy meeting at which Fed officials will update their economic and interest rate projections.

(Reporting By Karen Brettell Editing by Christina Fincher)