By Ann Saphir

(Reuters) -San Francisco Federal Reserve Bank President Mary Daly on Thursday said that recent cooler inflation readings are a "relief" and she expects further easing in both price pressures and the labor market to warrant interest rate cuts.

"With the information we have received today, which includes data on employment, inflation, GDP growth, and the outlook for the economy, I see it as likely that policy adjustments, some policy adjustments, will be warranted," Daly said in a group interview held by phone. "Exactly when that happens and when it would be appropriate to make an adjustment to policy is still unclear."

With inflation likely to cool further, though with potentially "bumpy" progress, the economy seems to be heading "more or less" to where one or two interest rate cuts this year as projected in the June Fed policymaker forecasts "would be the appropriate path," she said.

The Fed has kept its policy rate in the 5.25%-5.5% range since last July in an effort to cool inflation that by the consumer price index had peaked at 9.1% a year earlier.

Earlier on Thursday a government report showed the CPI slipped 0.1% last month after being unchanged in May, the weakest monthly reading since May of 2020, early in the pandemic.

The 3% year-over-year rise was the lowest reading in a year, helped by lower shelter-price inflation, a "welcome relief" Daly said and a trend she expects has more room to run.

A report last week showed the unemployment rate in June rose to 4.1% and job growth slowed - pointing to what Daly said was a cooling but still-solid labor market.

Daly did not rule out a move at the Fed's July 30-31 meeting, saying that every meeting is live and that she'll get more data before then from meetings with businesses, workers and community groups that will inform her vote.

The potential for inflation to accelerate, she said, has diminished since earlier in the year, though it still could be sticky, and it's unlikely the Fed will need to cut rates as rapidly as it raised them.

The question, she said, is no longer whether policy is restrictive - there is no evidence that it's not, she said - but rather "when do we loosen the reins."

And, she said, "it's a fairly big communications signal that you hear so many of us now, and Chair Powell importantly, talking about how important the labor market is," and no longer only about the need to bring down inflation.

The Fed needs to act before a sharp rise in unemployment, and before inflation reaches the Fed 2% goal, she said, or risk unnecessarily hurting workers and the economy.

Financial markets after Thursday's inflation data reflect expectations the Fed will start cutting rates in September and could deliver two more rate cuts by year's end.

(Reporting by Ann Saphir; Editing by Chizu Nomiyama and Andrea Ricci)