At 0818 GMT, the rouble was 0.3pc weaker against the dollar at 78.49 and had lost 0.9pc to trade at 85.84 versus the euro.
Brent crude oil, a global benchmark for Russia's main export, was down 2.9pc at $26.6 a barrel.
The US Senate backed a $2 trillion bill on Wednesday aimed at helping unemployed workers, but analysts feared a sharp uptick in the number of those without work in the world's largest economy, expected at 1230 GMT, would cause panic on global markets.
"Market sentiment was worrisome on the background of coronavirus spreading in the United States and also expectations of a significant increase in the number of weekly claims for unemployment benefits," said Veles Capital analyst Elena Kozhukhova.
Recorded coronavirus cases are beginning to rise more quickly in Russia, although the total is still far below the numbers in the United States and much of Europe.
The country said it would ground international flights from Friday and the mayor of Moscow said all restaurants, cafes, bars and shops, apart from those selling food and medicines, would close from March 28 until April 5.
Russia's budget can support new fiscal expenditures proposed on Wednesday, Sofya Donets, chief economist at Renaissance Capital said, adding that growth would be hit if a newly-announced week-long holiday of workers was prolonged.
"We calculated our tentative scenario for this new reality with both domestic and external restrictions to be prolonged, and we see -0.8pc y-o-y growth in Russia in 2020 in this scenario," she said.
Russian stock indexes were down.
The dollar-denominated RTS index was down 1.1pc to 970.7 points.
The rouble-based MOEX Russian index was 1.4pc lower at 2,417.9 points.