At 0743 GMT, the rouble was 0.3 percent weaker against the dollar at 62.18 and had lost 0.1 percent to trade at 69.10 versus the euro.
The dollar index, which tracks the U.S. currency against a basket of major rivals, added 0.3 percent to hit an eleven-week high.
Market participants are pricing in around a 70 percent chance that the Fed will raise rates in December, according to CME Group's FedWatch program, up from around 50 percent a few weeks ago.
Higher U.S. interest rates are seen pressuring emerging-market assets such as those denominated in roubles.
Brent crude oil, a global benchmark for Russia's main export, was little changed at 0743 GMT at around $53.1 a barrel.
Oil prices rose sharply on Monday, lifting Russian assets, after Russian President Vladimir Putin said Russia was ready to join OPEC countries in curbing output to rein in global crude oversupply.
Russian share indexes were mixed on Tuesday.
The dollar-denominated RTS index was down 0.2 percent to 1,015 points, while the rouble-based MICEX was 0.2 percent higher at 2,005 points.