Chicago Federal Reserve President Austan Goolsbee expressed caution Tuesday about lowering interest rates as the U.S. economy grapples with the forces of slower growth and a weaker labor market.

While he joined the rest of the Federal Open Market Committee last week in voting to cut the central bank's key borrowing rate , he told CNBC that further moves would depend on economic progress.

"I'm OK with moving to be in a better spot, and I think eventually, at a gradual pace, rates can come down a fair amount if we can get this stagflationary dust out of the air," he said during a " Squawk Box " interview. "But with inflation having been over the target for 4½ years in a row and rising, I think we need to be a little careful with getting all really up-front aggressive."

The FOMC voted

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