NEW YORK, June 24 (Reuters) - U.S. Treasury yields were largely unchanged on Monday as investors awaited economic and inflation data later this week to assess whether a recent weakening in economic activity will continue, which would strengthen the case for a first interest rate cut by the Federal Reserve in the coming months.

Yields, which move inversely to prices, have declined this month as price pressures have eased and data across different sectors of the economy, including the labor market, started to show moderation.

Benchmark 10-year yields are down by over 25 basis points so far this month and were at about 4.251% on Monday, just slightly lower than Friday. Two-year yields , which tend to more closely reflect monetary policy expectations, were last at 4.738%, marginally higher than their close last week.

Barring a rebound in inflation or a sharper-than-anticipated slowdown in the economy - both seen as unlikely scenarios at this stage - many in the market expect yields to move sideways until there is more clarity over the extent of any rate cuts.

"Markets are thinking the Fed is going to cut at some point ... but it is going to take a lot of damage to the economy to get rates to move much lower," said John Luke Tyner, head of fixed income and portfolio manager at Aptus Capital Advisors.

Investors will be looking at first quarter gross domestic product estimates released on Thursday and, more importantly, May inflation data on Friday, to get more clues about the U.S. central bank's next steps as it tries to battle inflation without causing a recession.

"I think there will be some wait and see in the market. PCE (personal consumption expenditures) inflation on Friday will be the biggest data release of the week," said Mona Mahajan, senior investment strategist at Edward Jones. "Otherwise a relatively quieter week and looks like that's been reflected in markets."

Federal Reserve Bank of Chicago President Austan Goolsbee said in a CNBC interview on Monday he was still looking for inflation to cool further as part of the process that would open the door to a rate cut.

Cleveland Fed President Loretta Mester, who is set to retire at the end of the month, told Reuters she expects inflation will cool over time and eventually allow the Fed to cut rates. She said she'd like to see "a few more months of data" before gaining confidence that easier policy is warranted.

Traders in futures tied to the Fed's policy rate were assigning a 61.2% chance to a first 25 basis point rate cut in September, LSEG data showed on Monday, with a total of nearly two rate cuts priced for this year.

Meanwhile, the gap between two- and ten-year yields remained deeply negative at about minus 48.6 basis points. An inversion in that part of the yield curve, which occurs when shorter-dated Treasuries yield more than longer-dated ones, has historically indicated that a recession is on the horizon.

(Reporting by Davide Barbuscia, Editing by William Maclean and Nick Zieminski )