FRANKFURT, May 19 (Reuters) - Incoming Federal Reserve Chair Kevin Warsh's suggestion that independence may not extend fully to the Fed's crisis-fighting role abroad has unsettled central banking peers, who fear any reduction in its global footprint could risk market stability.

With the dollar by far the world's most used currency, the U.S. central bank plays a pivotal role in stabilising financial markets during periods of stress. It has expanded its crisis-fighting tools over time to keep funding flowing.

But Warsh - U.S. President Donald Trump's pick for the job - raised eyebrows by suggesting that outside monetary policy, including in international finance, the Fed needs to work closely with the presidential administration and Congress.

Warsh told his confirmation hearing that independence in setting interest rates did not fully extend to the Fed's broader operations, prompting some to question whether it would remain fast and decisive when the next crisis hits.

U.S. A KEY BENEFACTOR

Warsh is expected to be sworn in soon by Trump, but no date has been announced. The Fed board said on Friday it had named Jerome Powell as chair pro tempore.

On- and off-the-record comments from more than half a dozen policymakers indicate they are attentive to Warsh's remarks and await clarification. But they anticipate no big policy change for now, if only because Fed liquidity facilities ultimately protect the U.S. economy as much as those of global partners.

A less reliable Fed would encourage countries to keep moving away from the dollar, extending and probably accelerating a 15-year fall in the greenback's global market share, they said.

There is little central banks can do in the short term if the Fed curbs access to dollars, however, and even the suggestion that liquidity lines may not be readily available could generate market turbulence.

"It's a double-edged sword," said one European Central Bank policymaker, who declined to be named. "The world relies on the dollar and if the dollar is not readily available, everybody pays a price - the U.S. included."

The Fed currently provides dollars on demand to the ECB and the central banks of Canada, Japan, Britain and Switzerland, against collateral, via standing liquidity tools. Other central banks can also access dollars through a more onerous facility.

The justification for that backstop is that commercial banks overseas sit on trillions of dollars worth of U.S. Treasury bonds, and that market stress could force them to sell quickly to access cash, importing turbulence to the United States.

Inserting politics into the provision of dollars would not be new. The Trump administration gave Argentina a $20 billion liquidity line ahead of elections last year, and Gulf and Asian nations recently requested liquidity lines to help deal with energy shocks and the fallout from the Iran war.

South Korean President Lee Jae Myung reportedly raised the issue during a meeting with U.S. Treasury Secretary Scott Bessent this month.

Nomura Research Institute economist Takahide Kiuchi, a former BOJ board member, noted the impact Fed policies can have on other markets.

"Warsh could attempt a tightrope of conducting dovish interest rate policy that aligns with Trump's hopes, while guiding a hawkish balance sheet policy," Kiuchi said.

"Any rupture in the U.S. market caused by such Fed moves, coupled with rising oil prices from the Iran war, could further push up 10-year Japanese Government Bond yields. That, in turn, could hurt Japan's economy and stock prices."

EURO NOT READY TO BENEFIT

Another source argued that it was in the Fed's interest to keep dollars flowing from abroad to help finance the high U.S. budget deficit.

A less reliable Fed could ultimately channel market demand towards the euro, the world's second most-traded currency. The ECB has made efforts to increase its market share and widen euro availability.

But the single currency's architecture is not yet adequate to take on a significantly greater role and vast internal reforms are still needed, a third source said.

All the sources agreed that contingency planning could help central banks live with a curtailed backstop, but that in times of crisis, the Fed remained the dollar lender of last resort.

"There isn't much you can do" to circumvent that, said Spyros Andreopoulos, founder of the Thin Ice Macroeconomics consultancy. "With the eurodollar market at 30 trillion dollars, by definition there aren't enough dollars in the system."

VETERANS DON'T MAKE RADICAL CHANGE

Many of the sources argued Warsh was unlikely to bring fundamental change, as he is an experienced central banker with a deep understanding of the Fed's core responsibilities.

"His comments were aimed at Trump and not so much at European counterparts," ING economist Carsten Brzeski said.

"Warsh is a Fed veteran, he's a financial crisis veteran, so he's very well aware of the potential financial stability issues if these swap lines were cut."

Others hoped Fed policymakers collectively would retain the safety net. Warsh, after all, has only a single vote and no one has questioned liquidity lines in the past.

"I worked with him during the financial crisis of 2008," Bank of Canada Governor Tiff Macklem said. "I believe that the culture and the comportment of the Fed will continue as it has in the past."

(Additional reporting by Leika Kihara in Tokyo and Promit Mukherjee in Ottawa; Editing by Mark John and Catherine Evans)