WASHINGTON, July 6 (Reuters) - U.S. Federal Reserve Governor Christopher Waller said on Monday that high inflation is the chief risk facing the Fed given a labor market that remains stable.

"A year ago I was advocating for rate cuts because the labor market was not looking good, so I was willing to tolerate a longer movement back to our 2% (inflation) target based on the labor market," Waller said at an economics conference in Rome. "Those risks have completely flipped around now. The labor market seems to be stabilizing in the US. Inflation has been taking off. So then that changes how you might want to think about policy."

Waller did not explicitly mention a June jobs report that showed weaker-than-expected hiring but a drop in the unemployment rate to 4.2% versus 4.3% in May. 

Yet his comments at an economics conference in Rome put added focus on inflation data scheduled to be released on July 14 on consumer prices through June, a key final data point before the Fed's July 28 to 29 meeting.

Global oil prices that have fallen back to around $70 a barrel, where they were before the start of a U.S.-backed war with Iran, could help ease headline inflation. But Fed policymakers in projections issued after the June meeting still felt their preferred measure of inflation would be more than a percentage point above the central bank's 2% target at year end. 

Investors currently expect rates to rise by the Fed's September meeting, with odds of a July rate hike at about one in four.

Whether it happens or not, with nine Fed officials projecting the need for tighter policy this year, "a rate hike is on the table" when policymakers gather for the July meeting, Tim Duy, chief U.S. economist for SGH Macro Advisors, wrote in a recent analysis. With relatively low unemployment and inflation stuck above target, "the Fed is missing on only one side of its mandate. This shouldn't be a debate anymore."

The Fed in a unanimous decision held rates steady at its June meeting, the first under new Chairman Kevin Warsh.