Cleveland Federal Reserve President Beth Hammack on Monday said the U.S. central bank faces challenges as it attempts to balance fighting stubborn inflation or protecting jobs.

"On the inflation side right now, I continue to be worried about where we are from an inflation perspective," Hammack told CNBC's "Squawk Box Europe."

"We have been missing our mandate on the inflation side, our objective of 2%, for more than four-and-a-half years and I continue to see that we have pressure in inflation both in the headline, in the core, and particularly, where I am worried about it, is I'm seeing it in the services," she added.

Asked whether it is mistake for the Federal Reserve to be cutting interest rates given the economic backdrop, Hammack described it as "a challenging time for monetary policy," saying the U.S. central bank was facing pressure on both sides of its mandate.

Her comments come shortly after stronger-than-expected economic data appear to have dented Wall Street's hopes for sharp monetary easing.

The Fed approved a widely anticipated rate cut earlier this month, lowering its benchmark overnight lending rate by a quarter percentage point to a range of 4.00%-4.25%, and signaled two more were on the way before the end of the year.

A robust batch of economic data since, however, has prompted investors to dial back their expectations for rapid rate cuts.

U.S. core inflation was little changed in August, according to data published Friday. The personal consumption expenditures price index posted a 0.3% gain for the month, putting the annual headline inflation rate at 2.7%, the Commerce Department reported late last week.

Excluding food and energy, the more closely followed core PCE price level was 2.9% on an annual basis after rising 0.2% for the month.

Hammack has previously suggested she would be hesitant about lowering interest rates as long as inflation remains a threat.

Indeed, more recently, Federal Reserve Chair Jerome Powell warned of a tricky path ahead on interest rates.

"Near-term risks to inflation are tilted to the upside and risks to employment to the downside — a challenging situation," Powell said on Sept. 23 during a speech to business leaders in Providence, Rhode Island.

"Two-sided risks mean that there is no risk-free path," he added.

— CNBC's Jeff Cox contributed to this report.