WASHINGTON (Reuters) - U.S. consumer prices fell for the first time in four years in June amid cheaper gasoline and moderating rents, firmly putting disinflation back on track and drawing the Federal Reserve another step closer to cutting interest rates in September.

The second straight month of benign consumer price readings reported by the Labor Department on Thursday should help to bolster confidence among officials at the U.S. central bank that inflation is cooling after surging in the first quarter.

The report also showed a measure of underlying inflation posting the smallest increase since August 2021 on a monthly basis. Financial markets saw a very high probability of the Fed starting its easing cycle in September.

"Barring rogue price data in July, the Fed has a checkered flag to reduce rates in September," said Brian Bethune, an economics professor at Boston College. "This guidance will be solidified at the July meeting."

The consumer price index dipped 0.1% last month, the first drop since May 2020, after being unchanged in May, the Labor Department's Bureau of Labor Statistics said. The CPI was weighed down by a 3.8% decline in gasoline prices, which followed a 3.6% decrease in May. Shelter costs, which include rents, increased a moderate 0.2% after advancing 0.4% in May.

Food prices rose 0.2% after edging up 0.1% in May. Grocery store prices ticked up 0.1%, with increases in dairy products, meat, fish and eggs offset by declines in the costs of fruits and vegetables as well as cereals.

In the 12 months through June, the CPI climbed 3.0%, the smallest gain since June 2023. That followed a 3.3% advance in May. Economists polled by Reuters had forecast the CPI ticking up 0.1% and gaining 3.1% year-on-year.

The broad moderation in inflation aligns with reports from retailers about consumers pushing back against higher prices. Retailers, including Target and Walmart, have cut prices on a range of goods. It was also a rare dose of good news for President Joe Biden in recent days, whose popularity has been eroded by the high cost of living.

The annual increase in consumer prices has slowed from a peak of 9.1% in June 2022. The CPI report followed news last week that the unemployment rate rose to a 2-1/2-year high of 4.1% in June from 4.0% in May.

Economic growth has also slowed in response to the central bank's hefty rate hikes in 2022 and 2023, with second-quarter gross domestic product forecast near the 1.8% annualized rate that policymakers view as the non-inflationary growth pace.

Fed Chair Jerome Powell has acknowledged the improving trend in price pressures, but told lawmakers this week he was not yet ready to declare inflation had been beaten and that "more good data" would strengthen the case for rate cuts.

Powell also highlighted risks to the labor market saying "we have seen considerable softening."

Economists said this suggested the Fed was shifting its focus to the labor market and could lower borrowing costs even if inflation remained above its 2% target. Financial markets saw a roughly 85% chance of a rate cut at the Fed's September meeting, compared with about a 70% chance seen before the report. Two rate cuts are anticipated this year.

"Today's report, and the subtle shift to a more balanced focus on slowing employment growth by the Fed, helps firmly put a September rate cut in the sights," said Richard de Chazal, macro analyst at William Blair.

A separate report from the Labor Department on Thursday showed initial claims for state unemployment benefits fell 17,000 to a seasonally adjusted 222,000 for the week ended July 6, the lowest level since late May.

The claims data included the Independence Day holiday. Claims tend to be volatile around holidays, and auto makers typically shut down assembly plants starting the July 4 week to retool for new models, injecting volatility in the data and making it harder to get a clean read of the labor market.

The number of people receiving benefits after an initial week of aid, a proxy for hiring, slipped 4,000 to a seasonally adjusted 1.852 million during the week ending June 29.

Stocks on Wall Street were mixed. U.S. Treasury yields fell. The dollar slipped against a basket of currencies.

BROAD PRICE MODERATION

The central bank has maintained its benchmark overnight interest rate in the current 5.25%-5.50% range since last July. It has hiked its policy rate by 525 basis points since 2022.

Excluding the volatile food and energy components, the CPI gained 0.1% in June. That was the smallest increase in the so-called core CPI since August 2021 and followed a 0.2% rise in May. The core CPI was restrained by a moderation in rents, which increased 0.3%, the smallest gain since August 2021.

Owners' equivalent rent (OER), a measure of the amount homeowners would pay to rent or would earn from renting their property, rose 0.3%. That was also the smallest increase since August 2021 and followed a 0.4% advance in May.

"This suggests that rental inflation may finally be breaking to the downside, which has been long anticipated given that real-time new rental agreements have pointed to more moderate increases," said Kathy Bostjancic, chief economist at Nationwide.

Consumers also got relief from healthcare costs, which rose 0.2% after advancing 0.5% in May. Airline fares were cheaper as were used cars and trucks, new motor vehicles and communication services. But motor vehicle insurance prices rebounded 0.9% after falling 0.1% in May.

Household furnishings and operations cost more as did personal care, education, recreation and apparel.

In the 12 months through June, the core CPI increased 3.3%. That was the smallest year-on-year advance since April 2021 and followed a 3.4% rise in May. Over the past three months, the core CPI increased at a 2.1% annualized rate, the smallest rise since March 2021 and a slowdown from 3.3% pace in May.

The moderation in CPI data was likely to be mirrored to some extent in the Personal Consumption Expenditures (PCE) price indexes. The PCE price indexes are the inflation measures tracked by the central bank for monetary policy.

Estimates for June core PCE inflation ranged from 0.13% to 0.19%. The core PCE price index edged up 0.1% in May. Core inflation was seen increasing 2.5% year-on-year in June after rising 2.6% in May. These forecasts can change after June's producer price report on Friday.

"There is light at the end of the tunnel finally after the central bank's long battle with inflation and interest rate cuts, lots of them, are on the way," said Christopher Rupkey, chief economist at FWDBONDS.