The benchmark S&P/ASX200 index on Monday finished down 9.7 points, or 0.11 per cent, to 8,570.4, while the broader All Ordinaries lost 5.0 points, or 0.06 per cent, to 8,815.3.
Over the weekend Mr Trump posted on social media that he would be imposing 30 per cent tariffs on the European Union and Mexico beginning August 1.
Leaders of the EU and Mexico said they'd keep negotiating with Mr Trump, so the threat might just be a negotiating tactic.
"We still think that major trading partners will negotiate their way out of Aug 1st tariffs and it is unlikely that tariffs will be as high as the latest round of letters suggest," J.P. Morgan's global market intelligence team wrote in a note.
"We still sit in the constructive camp."
On a six-day tour of China, meanwhile, Australian Prime Minister Anthony Albanese focused on developing trade and business ties between the two nations, speaking in Shanghai to business leaders from Rio Tinto, BHP and Fortescue as well as from Chinese steelmakers.
Four of the ASX's 11 sectors finished higher and seven closed lower, with energy and materials both gaining 0.5 per cent.
BHP rose 0.9 per cent to $39.73 and Rio Tinto added 0.6 per cent to $111.74 while Fortescue dipped 0.5 per cent to $16.90.
South32 fell 5.1 per cent to $2.95 after the BHP spinoff announced that it expected to write off some of the value of its Mozal aluminium smelter in Mozambique, as it had been unable to source affordable electricity for it.
Goldminers were higher as the yellow metal traded at three-week high of $US3,383 an ounce.
Northern Star rose 1.7 per cent, Evolution climbed 1.9 per cent and Vault Minerals had advanced 2.6 per cent.
Rare earth miner Lynas climbed 3.4 per cent to a $10 following the Pentagon's decision last week to take a billion-dollar stake in a US rare earth miner.
In the financial sector, NAB was up 0.1 per cent to $39.66 but the other three big banks were lower. CBA dropped 0.4 per cent to $178.72, ANZ retreated 0.8 per cent to $30.08 and Westpac slipped 0.7 per cent to $33.56.
Elsewhere, Droneshield soared 17 per cent to an all-time high of $3.24 after the military contractor said it would spend $13 million to build a new production facility in the Sydney suburb of Alexandria to meet high demand for its anti-drone products.
Hansen Technologies advanced 10.9 per cent to a two-month high of $5.30 after the software provider to the energy and utility companies upgraded guidance, citing disciplined cost management.
Abacus Storage King gained 5.8 per cent to $1.56 after the self-storage giant said a consortium looking to acquire it had lifted its tentative takeover offer by nearly 15 per cent, to $2.17 billiion.
The offer is being made by New York-listed Public Storage and Ki Corporation, the family office of South African businessman Nathan Kirsh.
The Australian dollar was buying 65.72 US cents, unchanged from close of business on Friday.
Bitcoin had set another all-time high, at $US122,500, or more than $A186,000 on Australian exchanges.
The original cryptocurrency is up 3.7 per cent in the past 24 hours and has roughly doubled in value over the past 12 months.
ON THE ASX:
* The benchmark S&P/ASX200 index finished Monday down 9.7 points, or 0.11 per cent, at 8,570.4
* The broader All Ordinaries dropped 5.0 points, or 0.06 per cent, to 8,815.3.
CURRENCY SNAPSHOT:
One Australian dollar buys:
* 65.72 US cents, from 65.72 US cents at 5pm on Friday
* 94.80 Japanese yen, from 94.58 Japanese yen
* 55.79 euro cents, from 55.75 euro cents
* 48.05 British pence, from 48.14 pence
* 108.17 NZ cents, from 108.21 NZ cents
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