SINGAPORE, April 15 (Reuters) - Asia shares edged slightly higher on Tuesday, helped by gains in auto companies after U.S. President Donald Trump suggested he might grant exemptions on auto-related levies already in place.

U.S. Treasury bonds steadied having staged a recovery overnight following last week's historic selloff, while the dollar continued to fall out of favour with investors.

Trump said on Monday he was considering a modification to the 25% tariffs imposed on foreign auto and auto parts imports from Mexico, Canada and other places. Those tariffs could raise the costs of a car by thousands of dollars, and Trump said car companies "need a little bit of time because they're going to make 'em here."

That followed Friday's move to exempt smartphones, computers and some other electronics from Trump's "reciprocal" U.S. tariffs. But his administration later stepped up probes into imports of semiconductors after Trump said on Sunday he would be announcing their tariff rate over the next week.

"When we start to see some of these exemptions flow through for particular sectors, it helps markets think about tariffs as something that aren't necessarily going to be all-encompassing, and that they might actually be reprieved," said Illiana Jain, an economist at Westpac.

Investors took whatever good news they could get after last week's heavy selling across markets and pushed shares slightly higher. MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.3%.

Japan's Nikkei rose 1%, with shares of auto companies like Toyota ( TM ) and auto parts maker Denso ( DNZOF ) among the top gainers on the index.

But gains were limited 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.

U.S. futures swung between losses and gains to last trade lower after an overnight gain on Wall Street.

Nasdaq futures and S&P 500 futures were each down close to 0.2%. In Europe, EUROSTOXX 50 futures fell 0.14%, while FTSE futures were up 0.25%.

Investors have more earnings to weather this week with Bank of America and Citigroup among the big banks reporting. Numbers from chipmaker TSMC later in the week will also be a highlight.

China's CSI300 blue-chip index and Shanghai Composite Index eased more than 0.4% each, while Hong Kong's Hang Seng Index reversed early gains to fall 0.16%.

"At the margin, the uncertainty and re-ordering of the global trade system is both inflationary and suggestive of slower growth," said Bharat Sachanandani, head of flow strategy and solutions for Asia Pacific at Societe Generale.

"Asset markets appear to tell us that higher prices for U.S. consumers will be met with demand destruction, and probabilities of recession are rising."

U.S. RATES

U.S. Treasuries held onto overnight 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 was steady at 4.3564%, after having fallen nearly 13 basis points in the previous session.

Similarly, the two-year yield was little changed at 3.8450% after sliding 12 bps on Monday.

Some analysts said comments from Federal Reserve Governor Christopher Waller also contributed to the fall in yields.

He said on Monday that the Trump administration's tariff policies are a major shock to the U.S. economy that could lead the Fed to cut rates to head off recession even if inflation remains 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 easing by December, with most expecting the Fed to hold rates next month.

In currencies, the dollar held near a three-year trough against the euro at $1.13245 and was not far from its decade-low against the Swiss franc.

"(The) behaviour of the U.S. dollar recently has changed - it is now ignoring rate differentials, and responding more to capital flows," said SocGen's Sachanandani.

"The U.S. dollar is not liking the prospect of U.S. corporations being less profitable, U.S. consumers facing higher inflation and foreign investors having a collapsing appetite for U.S. assets."

Oil prices rose, boosted by the latest tariff exemptions floated by Trump. Brent crude futures gained 0.2% to $65.01 per barrel while U.S. crude was up 0.24% to $61.68.

Spot gold held near a record high at $3,221.45 an ounce.

(Reporting by Rae Wee; Editing by Kim Coghill)