The Federal Reserve needs to keep raising borrowing costs to bring high inflation under control, a string of U.S. central bank officials said on Thursday, even as they debated how fast and how high to lift them.

St. Louis Fed President James Bullard said that given the strength of the economy he is currently leaning toward supporting a third straight 75-basis-point interest rate hike in September so as to drive down inflation more quickly.

"I don't really see why you want to drag out interest rate increases into next year," Bullard told the Wall Street Journal, saying he would like to get the Fed's benchmark overnight interest rate to a target range of 3.75% to 4.00% by the end of this year. The Fed's policy rate is currently 2.25%-2.50%.

Earlier on Thursday, San Francisco Fed President Mary Daly said hiking rates by 50 or 75 basis points next month would be a "reasonable" way to get short-term borrowing costs to "a little bit above" 3% by the end of this year, and on their way to a little bit higher in 2023.

The exact pace of the rate hikes would depend on employment data, which has shown brisk growth in recent months, and inflation, Daly told CNN International. Inflation, by the Fed's preferred measure, is running at more than three times the central bank's 2% target.

And with the global economic slowdown acting as a headwind on U.S. growth, she said "we have to take that into consideration as we ensure that we don't overdo policy."

The two Fed officials' views suggest a split in the Fed between those who want to push rates higher quickly, and those who are more cautious because of potential damage to the job market and the risk of a rise in the U.S. unemployment rate, now at 3.5%.

But Bullard and Daly were clear that after rates have risen to a certain level, the Fed will not be quick to cut them. Bullard said market expectations of rate cuts were "definitely premature." Daly said she supported a "raise-and-hold" strategy.

"The worst thing you can have as a business or a consumer is to have rates go up and then come rapidly down ... it just causes a lot of caution and uncertainty," Daly said. "I do think we want to not have this idea that we'll have this large hump-shaped rate path where we'll ratchet up really rapidly this year and then cut aggressively next year - that's not what's on my mind."

Speaking at a separate event, Kansas City Fed President Esther George said she and her colleagues would continue to debate the question of how fast to raise rates, but that they would not stop until they are "completely convinced" that inflation is coming down.