WASHINGTON (Reuters) -Federal Reserve officials at their last meeting acknowledged the U.S. economy appeared to be slowing and that "price pressures were diminishing," but still counseled a wait-and-see approach before committing to interest rate cuts, according to minutes of the June 11-12 session.

The minutes, which were released on Wednesday, noted in particular a weak May reading in the consumer price index as one among "a number of developments in the product and labor markets" that supported a view that inflation was falling.

Wage growth had slowed, a few participants noted, while others pointed to price cutting among major retailers and reports from their own business contacts that "pricing power had declined."

But if the narrative around inflation pointed to faith that it was on the way down, U.S. central bank policymakers weren't ready, yet, to open the door to rate cuts.

Officials "did not expect that it would be appropriate to lower the target range for the federal funds rate until additional information had emerged to give them greater confidence that inflation was moving sustainably toward" the 2% target, the minutes said.

At the time of the meeting, the personal consumption expenditures price index, which is used to set the Fed's inflation target, had been reported at 2.7% on a year-on-year basis for the month of April.

Policymakers still judged that reading to be "elevated" and representing only "modest" improvement since their last meeting, a fact that warranted continued tight monetary policy even though the economy appeared to be slowing and price pressures waning, the minutes said.

The U.S. government reported last week that the PCE fell to 2.6% in May.

"The vast majority of participants assessed that growth in economic activity appeared to be gradually cooling, and most participants remarked that they viewed the current policy stance as restrictive," and therefore likely to further curb the economy and inflation, the minutes said.

But in voting to keep the policy rate steady in the 5.25%-5.50% range where it has now been for a year, "participants noted that progress in reducing inflation had been slower this year than they had expected last December," the minutes said, with "some participants" emphasizing the need for patience before cutting rates, and "several" citing the possible need to raise rates further if inflation resurged.

Yet the minutes "lean dovish," Michael Feroli, chief U.S. economist at J.P. Morgan, wrote in a note, with the emphasis on the number of factors aligning to lower inflation.

Ian Shepherdson, chief economist at Pantheon Macroeconomics, said the minutes reflected a central bank that was "undogmatic" in its view of the economy and seemingly poised to react fast to any coming turns in the data.

The Fed "remains in no rush to start easing, but minds will change quickly if, as we expect, employment growth slows and inflation continues to moderate," Shepherdson wrote, projecting the Fed will cut rates much faster than many expect, by a full 1.25 percentage points by the end of this year, if job growth declines as much as he anticipates.

Investors broadly expect a quarter-percentage-point rate cut at the Fed's Sept. 17-18 meeting and another one in December.

POLICY IN TRANSITION

The risk of a jobs slowdown also was noted in the minutes, as was the need for Fed officials to prepare the public and investors for a number of possible economic outcomes.

Many Fed officials have begun speaking in terms of alternate "scenarios" to frame their policy views, and the minutes put that approach at the center of the discussion.

"Participants emphasized the importance of conditioning future policy decisions on incoming data, the evolving outlook, and the balance of risks," the minutes noted, and downplaying any sense that decisions were on a "preset path."

Among those possibilities, increasingly, is the risk that the job market might slow much faster than anticipated.

"Several participants specifically emphasized that with the labor market normalizing, a further weakening of demand may not generate a larger unemployment response than in the recent past when lower demand for labor was felt relatively more through fewer job openings," the minutes said, citing an argument that has been sketched out in detail by Fed Governor Christopher Waller.

After higher-than-anticipated inflation early this year raised fears the Fed might again be a step behind in its battle against rising prices, the minutes seemed to confirm the transition towards looser monetary policy remains underway, but cautiously so.

Data released on June 12 showed the CPI had not risen at all in May on a month-to-month basis, an encouraging development that came late in the Fed's policy deliberations.

Some market players had been surprised the more favorable data was not reflected more fully in the Fed forecasts released at last month's meeting.

The U.S. central bank will hold its next policy meeting on July 30-31, when it is expected to leave its benchmark interest rate unchanged.

Policymakers by then will get an update on the labor market with the release on Friday of the employment report for June, the release of the CPI for June on July 11 and an initial estimate of second-quarter economic growth on July 25.