At 0723 GMT, the rouble was 0.2pc weaker against the dollar at 64.09 and had lost 0.3pc to trade at 70.53 versus the euro.
Global market activity is expected to be weaker on Thursday due to the Thanksgiving Day holiday in the United States.
Trump's move drew the ire of Beijing, with which he is seeking a deal to end a damaging trade war between the world's two largest economies.
Brent crude oil, a global benchmark for Russia's main export, was down 0.3pc at $63.9 a barrel.
Russian stock indexes were down, with business conglomerate AFK Sistema continuing to lose ground in Moscow, after the company posted mixed third-quarter results on Wednesday.
The dollar-denominated RTS index was down 0.1pc to 1,439.6 points. The rouble-based MOEX Russian index was narrowly down at 2,928.2 points.