At 0800 GMT, the rouble was 0.15 percent weaker against the dollar at 61.72 and had lost 0.21 percent to trade at 75.22 versus the euro.
The dollar inched up, approaching its recent four-month high, as the US 10-year bond yield poked above 3 percent to hit its highest level since early 2014.
"The fundamental reason for such a move (in UST) remains the same: the expectation of a more rapid normalization of the Federal Reserve policy", analysts at Rosbank, a Russian unit of Societe Generale, said in a note.
The rouble drew some support from high oil prices, which reached more than three-year highs in the previous session.
Brent crude oil, a global benchmark for Russia's main export, was up 0.1 percent at $73.92 a barrel.
The rouble also benefited from month-end tax payments that usually prompt export-focused companies to convert their dollar revenues to meet local liabilities.
Russian stock indexes were down as the main US indexes fell back yesterday.
The dollar-denominated RTS index was down 0.63 percent to 1,146 points. The rouble-based MOEX Russian index was 0.37 percent lower at 2,245 points.