In morning trade, Japan’s Nikkei 225 index was about 0.4% higher, South Korea’s Kospi was up 0.6% and stocks on Australia’s ASX 200 rose by around 1%.
Meanwhile, the dollar was lower against some other major currencies, including the pound and the euro.
US markets were closed on Monday for the Martin Luther King Jr Day holiday.
Trump has not yet enacted any new tariffs but said on Monday that he is considering imposing 25% import taxes on Mexico and Canada on 1 February.
During the election campaign, Trump also pledged tariffs of 10% on all imports into the US and said he would hit China with a 60% import tax.
He has said tariffs will make Americans richer, although critics say the costs are likely to be passed on to consumers.
The President has also said he would create an “External Revenue Service” to collect all tariffs, duties and revenues from foreign sources.
