Investors sent a warning Wednesday that central-bank independence matters to Wall Street.

Reports that President Trump was again pondering firing Federal Reserve Chairman Jerome Powell drove longer-term Treasury yields higher and the dollar lower. The markets see the prospect of higher inflation if a new Fed regime were to cut rates in line with Trump’s wishes. Some analysts say there would be global consequences, because Treasurys and the dollar underpin financial markets worldwide.

Trump told reporters Wednesday at the White House that it was “highly unlikely” he would fire Powell, adding later, “Unless he has to leave for fraud.” Trump alluded to an effort by White House advisers to target renovations at Fed properties that have cost more than planned.

The market’s moves faded as the day went on, but the message was clear: Don’t mess with the Fed.

Past examples of tension between heads of government and their central bankers offer clues about how the Trump-Powell showdown could play out. Politicians usually want lower rates, mindful of consumer and business borrowing costs. Central bankers need to promote stable prices.

“Political pressure and in this case, presidential pressure, increases the risk that there will be a pedal-to-the-metal inflationary episode,” said Mark Spindel, an investment manager who co-wrote a history of Fed independence.

‘Squeezing blood’

The modern concept of Fed independence can be traced to the 1951 Treasury-Fed accord, which gave the central bank a freer hand to set interest rates as it saw fit.

The Fed since has enjoyed considerable autonomy, though it hasn’t been immune to political pressure. Former President Lyndon Johnson urged Fed leader William McChesney Martin Jr. to keep rates low while he pursued his broad social-spending agenda.

In an October 1965 meeting, Johnson stretched out his arm and clenched his fingers in describing how a rate increase would “amount to squeezing blood from the American working man in the interest of Wall Street,” Paul Volcker, then an assistant Treasury secretary, recalled in his memoir.

When Martin cast the deciding vote to raise rates two months later, he warned colleagues they were risking a furious backlash from Johnson. Martin was summoned to Johnson’s Texas ranch days later, where the president reportedly yelled at him.

Burns vs. Nixon

The next Fed chair, Arthur Burns, was a well-regarded economist who was known as an inflation fighter when former President Richard Nixon appointed him to the post in 1970. But historians say Burns became overly preoccupied with currying favor with the president, who desperately wanted to hold rates low ahead of his 1972 re-election campaign, despite growing inflation.

“Evidence from the Nixon tapes clearly reveals that President Nixon pressured Burns, both directly and indirectly…to engage in expansionary monetary policies prior to the 1972 election,” researchers wrote in a 2006 paper in the Journal of Economic Perspectives.

Inflation was out of control by the late 1970s, stoked in part by Burns’s reluctance to raise rates early on in the decade. In the aftermath, former President Ronald Reagan and some other future occupants of the White House refrained from publicly criticizing the Fed.

Bank of England independence

Central-bank watchers point to the Bank of England’s more recent independence as a success story. Founded in 1694, the world’s second oldest central bank had longstanding reporting lines into the monarchy and Her Majesty’s Treasury, the U.K.’s economic and finance ministry.

“Elections influenced the nature and timing of decisions on interest rates,” former Bank of England governor Mervyn King wrote in 2017. “When I arrived at the BoE as chief economist in 1991, it was clear that interest rates could change at any time on any day.”

By the early 1990s, the pound was increasingly volatile, heavy currency intervention wasn’t working and a panicked government lost credibility. The pound crashed against the dollar in 1992, disrupting the economy.

In an effort to improve investor confidence, Britain’s new Labour government in 1997 granted the central bank the ability to set interest rates independently. British stocks soared, and government bond yields fell in the following years.

“It was a more progressive, left wing administration, but the move enabled them to distance themselves from some of these decisions,” Spindel said “The stock and bond markets afforded a huge vote of confidence when they took away the prime minister’s pressure.”

Turkish turmoil

In a more recent—and extreme—example, Turkish President Recep Tayyip Erdogan sacked his country’s central banker in early 2021. As the developing economy struggled with the twin specters of slowing growth and persistent inflation, Erdogan fired central bank governor Naci Agbal after Agbal raised interest rates to rein in rising prices.

Erdogan claimed that lowering interest rates would reduce inflation by cutting corporate borrowing costs. The idea contradicted widely accepted economic principles that low interest rates push prices up by spurring economic activity.

“The predictable result was blowup inflation and eventually a move toward monetary orthodoxy on the part of Erdogan,” said Kasper Bartholdy, an economist who worked in emerging markets fixed income strategy for Credit Suisse at the time.

Turkey’s inflation more than tripled to 72% in 2022 and the lira lost about 60% of its value against the U.S. dollar, fueling economic pain and popular discontent. Fitch Ratings slashed Turkish government bonds to single-B in 2022 before Erdogan appointed an economic team in 2023 that raised interest rates.

Write to Jack Pitcher at jack.pitcher@wsj.com and Matt Wirz at matthieu.wirz@wsj.com