A September interest rate hike is "very much in play" if the U.S. economy continues to strengthen, though the Federal Reserve could also wait until December to start tightening policy, an influential Fed official said in a newspaper interview.

New York Fed President William Dudley, quoted in the Financial Times, said, "It would not shock me if we decided to lift off in September, or it wouldn't shock me if the data were a little softer and it caused us to wait."

Dudley, a dovish policymaker who is a close ally of Fed Chair Janet Yellen, was said to cite recent U.S. wage gains, income, and household spending as having improved his outlook for the economy.

Greece, which is in a tense standoff with its European creditors, remains a "huge wild card," he said.

"If we hit 2.5 percent growth in the second quarter and it looks like the third quarter is shaping up for something similar, then I think you are on a firm enough track that you would imagine you would have made sufficient progress" in the Fed's inflation and employment goals to raise rates "certainly by the end of the year," Dudley was quoted as saying.