WASHINGTON, July 14 (Reuters) - U.S. Federal Reserve Chairman Kevin Warsh on Tuesday vowed to "do my job" if challenged by President Donald Trump, his most direct comment so far about how he would handle the sort of pressure his predecessor felt through much of his tenure.

Asked about how he will react if Trump continues targeting the U.S. central bank through efforts that included the attempted firing of Fed Governor Lisa Cook, Warsh told the House of Representatives Financial Services Committee that the U.S. Supreme Court had recently reaffirmed the Fed's independence in setting monetary policy.

If targeted personally, "I would continue to do my job," Warsh told lawmakers, echoing comments former Fed Chair Jerome Powell made at various points when asked what he would do if Trump tried to fire him. "Outside the four walls of the Federal Reserve there's no doubt a lot of politics. My goal inside the central bank is for there to be no politics. To the extent there's politics there, we're going to get rid of it."

"The independence of the Fed is sacrosanct," Warsh said in response to one of several questions about his willingness to set policy based on data even if the president was pushing for lower borrowing costs. "Credibility is bolstered if we are and are perceived to be independent. ... That is the way we can best do our job."

Warsh's relationship with Trump, who was effusive about his pick to lead the Fed at a swearing-in ceremony in late May, formed the backdrop of the first of Warsh's two days of testimony in Congress this week, with Democrats warning him not to rely solely on the recent Supreme Court ruling to defend the central bank's independence.

The new Fed chief, who took over from Powell, will be managing both the economy and Trump during a difficult-to-read moment. Inflation is currently running above the Fed's 2% target, and there is uncertainty around whether renewed conflict in the Middle East could reverse recent progress.

Data released earlier on Tuesday showed that U.S. consumer inflation slowed more than expected to 3.5% on a year-over-year basis in June on lower energy prices.

Warsh said he would not "cherry-pick" the moment as a sign of progress.

"There might be some that look at this morning's data and say, 'Oh, mission accomplished, everything is swell.' That is not my view," Warsh told lawmakers.

Traders see only about a 12% chance of a quarter-percentage-point rate increase at the Fed's July 28-29 meeting, versus about 42% on Monday, according to CME Group's FedWatch tool. They give a hike at the September 15-16 meeting about a 53% likelihood, down from about 75% on the prior day.

In his testimony, Warsh reiterated that his main thrust right now is to bring inflation back to that target, a goal welcome among those he called his "superb colleagues" at the central bank but that may require him to disappoint Trump's repeated hope for lower interest rates.

"If we get policy right - and we will - the inflation surge of the last five years will be a thing of the past," Warsh said.

Warsh will appear before the Senate Banking Committee on Wednesday, a panel that in late April recommended his confirmation as Fed chief on a party-line vote, with Democrats expressing particular concerns about his relationship with Trump and whether he would be truly independent after a selection process in which the president said he would only nominate someone he was certain would cut rates.

NO SIGNS OF IMMINENT RATE CUT

Warsh's initial steps are seen as showing more distance from Trump than not, with his appointments to a set of task forces last week noted for the level of expertise and the absence of the sort of ideological or partisan figures brought in at other agencies.

"If people were concerned he would be a 'sock puppet,' those fears should have been gone after the first press conference" following the Fed's decision to hold rates steady, when Warsh's comments were seen, if anything, as tilted toward keeping them that way, Jon Faust, a former top adviser to Powell and now an economics professor at Johns Hopkins University, said prior to Warsh's appearance on Tuesday.

The task force appointments "really cement that view," tapping a group of well-known economists, corporate executives and central bankers likely to operate largely as Warsh says he intends - as neutral arbiters of key debates, some new and some ongoing, Faust said.

Warsh has shown no signs that a rate cut will happen soon, a policy stance no different from the one that earned Powell steady criticism from Trump and, near the end of his tenure, a since-abandoned criminal investigation. Powell remains a Fed governor.

"I agree that he got the president's support by giving him dovish signals," said Samuel Tombs, chief U.S. economist at Pantheon Macroeconomics. "The counterpoint is that now that Warsh is in position, he has the luxury of taking a long, impartial view ... Powell successfully showed how independent a Fed chair can be, how political interference can only get so far. I'm sure Warsh has one eye on who Trump's successor might be," with his legacy and possible reappointment both at stake.

WARSH'S STANDING WITH TRUMP MAY BE TESTED

If anything, Warsh has begun talking in more conditional terms about some of the things, like artificial intelligence, that before his nomination he said could lower inflation and lead to lower rates.

The monetary policy report submitted to Congress last week ahead of his testimony noted that AI investment is pushing up some prices, and other Fed policymakers have noted how it may be feeding a jump in software costs that is adding inflationary pressure from a new direction.

Warsh has also acknowledged that the timing of supply-side and productivity gains from AI is uncertain, while the impact on demand for capital, skilled construction labor, and infrastructure is occurring now.

While Warsh, unlike his colleagues, did not submit an interest rate projection at the Fed's June 16-17 meeting and does not plan to do so, since he opposes that sort of "forward guidance," he said at his press conference that there was only a single policy proposal on the table at his first meeting as Fed chief - with no discussion of a rate cut.

Like others who started out on Trump's good side but later lost favor, Warsh's standing with the president could be tested in coming months - if high inflation persists, for example, and support among Warsh's colleagues for rate hikes becomes too great to ignore.

But for now, at least, Warsh seems to be taking Trump at his word when the president at his swearing-in told him to "be totally independent. ... Don't look at me."

"The president says he wants Kevin to do what he thinks is best. I don't know if you can say how long that will last," said former Cleveland Fed President Loretta Mester. But "so far so good," said Mester, who described the composition of the new Fed task forces as "very promising."