BOSTON/LONDON (Reuters) -Some trade policy relief and strong bank earnings were not enough to keep Wall Street from pushing U.S. stocks down slightly on Tuesday, although U.S. government bonds and the dollar regained some ground after sharp declines last week.

U.S. President Donald Trump on Monday said he was considering a modification to the 25% tariffs imposed on foreign auto and auto parts imports from Mexico, Canada and other places.

That followed the move late on Friday to exempt smartphones, computers and some other electronics from Trump's "reciprocal" tariffs. 

White House press secretary Karoline Leavitt said on Tuesday that Trump is open to making a trade deal with China but Beijing should make the first move.

The main U.S. stock indexes ticked lower on Tuesday, even as Bank of America, Citigroup and Wells Fargo gained after the trio of banking giants posted strong profits for the first quarter. The Dow Jones Industrial Average fell about 0.4%, the S&P 500 dipped about 0.2%, and the Nasdaq Composite was virtually flat.

"The market is eerily calm," investment strategist Louis Navellier wrote in a note on Tuesday. "It's a bit unnerving after the rollercoaster ride we've been on since the tariff tantrum began."

Outside the U.S., investors took whatever good news they could get after the recent heavy selling across markets and pushed shares higher. The pan-European STOXX 600 index rose 1.6% on Tuesday, led by the autos and parts sector whose gauge jumped about 2.3%. 

In Asia, MSCI's broadest index of Asia-Pacific shares outside Japan gained 1%. Japan's Nikkei rose 0.8%, with shares of auto companies like Toyota and auto parts maker Denso among the top gainers. 

Analysts remained cautious, however, as uncertainty over Trump's trade policies, and his constant back-and-forth on tariffs, continued to cast a cloud over markets and the global economic outlook.

Darrell Cronk, president of the Wells Fargo Investment Institute, wrote in a note on Tuesday that the "final tariff menu remains unsettled" and will decide if there is a recession or not.

"We should expect volatility to remain high, but last week proved the power of markets to push the administration not to break the financial system," Cronk added. "Hence, we should have a floor for equities and a ceiling for rates."

BOND YIELDS STEADY

U.S. Treasuries added to Monday's gains on Tuesday after a manic selloff last week that led to the largest weekly increase in borrowing costs in decades. Bond yields move inversely to prices.

The benchmark 10-year yield fell about 3 basis points to 4.333%, having fallen nearly 13 basis points in the previous session.

Federal Reserve Governor Christopher Waller said on Monday that the Trump administration's tariff policies were a major shock to the U.S. economy that could lead the Fed to cut rates to head off recession even if inflation remained high. Atlanta Fed Bank President Raphael Bostic, meanwhile, suggested the U.S. central bank should stay on hold until there is more clarity.

Markets are now pricing in about 85 bps worth of monetary policy easing by the end of the year, with most expecting the Fed to hold rates next month.

The dollar gained slightly on Tuesday, but still traded near a three-year low against the euro and a six-month trough against the yen, as investors trying to make sense of the constant changes to tariffs remained wary of U.S. assets. The dollar index, which measures the greenback against a basket of currencies including the yen and the euro, ticked up 0.3% on the day.

"The U.S. exceptionalism narrative that had previously underpinned the surge in U.S. equity markets over the past couple of years, and boosted the dollar, has lost much of its shine," said Jonas Goltermann, deputy chief markets economist at Capital Economics.

Oil prices were little changed, slipping 0.3%, as investors tried to figure how much the U.S.-China trade war could reduce global economic growth and oil demand. Brent crude futures settled 21 cents lower at $64.67 per barrel, while U.S. West Texas Intermediate (WTI) crude fell 20 cents to end at $61.33.

Gold prices rose 0.7% to $3,232 an ounce, helped by safe-haven demand, while the overall weaker dollar also lent support. [GOL/]

(Reporting by Lawrence Delevingne in Boston, Samuel Indyk in London and Rae Wee in Singapore; Editing by Marguerita Choy and Cynthia Osterman)