(Reuters) - A government report showing the U.S. labor market continues to cool could give Federal Reserve policymakers added confidence they are winning their fight on inflation, sparking a more active debate on interest-rate cuts at their next meeting.     The Labor Department's monthly report showed the unemployment rate rose in June to 4.1%, and big revisions to prior-month estimates meant the average monthly payroll gain over the most recent three months downshifted to 177,000.

That's below the 200,000-a-month gain that Fed Governor Lisa Cook recently estimated the economy now needs to create just to keep up with immigration and other increases to the population.

Average hourly earnings were up 3.9% from a year earlier, the report also showed, falling short of a 4% pace for the first time in three years and adding to evidence that price pressures may be easing.

U.S. central bankers meeting at the end of this month are not expected to change their policy rate from the 5.25%-5.5% range it has been in since last July. But the fresh data, along with other recent reports suggesting that inflation is cooling and the economy is slowing, could put a rate cut at the following meeting in their sights, analysts said.

"Overall, a moderation in payrolls in Q2 coupled with a rise in the unemployment rate and a slower growth path suggested by recent data bolster the case for rate cuts this year," said Rubeela Farooqi, Chief US Economist of High Frequency Economics. "We think the Fed could certainly start the discussion about cutting rates at the upcoming FOMC meeting, and lower the policy rate in September, if the data continue to show moderation."

Fed policymakers at their June meeting signaled they see just one interest-rate cut this year, a forecast that pointed to a December start to any policy easing. Fed Chair Jerome Powell said they would need to be confident inflation is heading to their 2% goal before cutting rates.

He also said any unexpected weakening in the labor market could also trigger a rate cut.

Powell is slated to address Congress next week, and investors will be watching keenly for how he views the latest data and what it means for the Fed's policy path.

Financial markets are pricing in a September rate cut, with the implied probability remaining at about 72% after the June jobs report. Traders are pricing in a second rate cut in December.

(Reporting by Ann Saphir, Howard Schneider, Michael S Derby; Editing by Andrew Heavens and Chizu Nomiyama)