NEW YORK, June 13 (Reuters) - U.S. Treasury yields were lower on Thursday after economic data showed a cooling of the labor market and price pressures, keeping hopes intact a rate cut from the Federal Reserve may be on the horizon.

The Labor Department said the producer price index (PPI) for final demand dropped 0.2% last month after advancing by an unrevised 0.5% in April, and below the 0.1% increase forecast by economists polled by Reuters. In the 12 months through May, the PPI increased 2.2% after rising 2.3% in April.

The data comes after a gauge of consumer prices (CPI) on Wednesday was unchanged in May for the first time in almost two years.

"When you look at today's PPI and yesterday's CPI, it's unambiguously good news on the inflation front. Of course, one report doesn't make a trend, but these are the sort of numbers and reports we need to see to be convinced that inflation is coming down, and for the Fed to be convinced that inflation is coming down," said Collin Martin, fixed income strategist at the Schwab Center for Financial Research in New York.

A separate reading of the labor market showed weekly initial jobless claims climbed 13,000 to a seasonally adjusted 242,000, a 10-month high, and above the 225,000 estimate.

"When you see the rise in initial jobless claims, that just suggests that the labor market is coming more into balance, and it suggests that growth should moderate a bit down the road," Martin said.

The yield on the benchmark U.S. 10-year Treasury note fell 4.3 basis points to 4.252%.

Yields had dropped after the consumer price index report on Wednesday but pared some declines after the Federal Reserve left interest rates unchanged and pushed out the start of rate cuts to perhaps as late as December.

The yield on the 30-year bond declined 3.6 basis points to 4.414%. A $22 billion auction in 30-year bonds was seen as strong by analysts, with the above-average demand of 2.49 times the notes on sale the highest in a year, according to LSEG data.

Market expectations for a rate cut of at least 25 basis points at the Fed's September meeting stand at 68.5%, according to CME's FedWatch Tool, up slightly from the 64.7% in the prior session.

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 negative 44.7 basis points.

The two-year U.S. Treasury yield, which typically moves in step with interest rate expectations, shed 5.3 basis points to 4.697%.

The breakeven rate on five-year U.S. Treasury Inflation-Protected Securities (TIPS) was last at 2.175% after closing at 2.19% on June 12.

The 10-year TIPS breakeven rate was last at 2.22%, indicating the market sees inflation averaging about 2.2% a year for the next decade.

(Reporting by Chuck Mikolajczak; Editing by Andrea Ricci and Nick Zieminski)