At 0733 GMT the rouble was around 1.1 percent weaker against the dollar at 58.57 and lost 0.7 percent to trade at 63.44 versus the euro.
Brent crude oil, an important driver for all Russian assets, was trading 1 percent lower at around $55.80. The dollar was trading 0.5 percent up against a basket of major currencies .
Russia's monthly tax period, during which companies convert foreign currency to pay taxes to the state budget, ends on Monday, removing an important pillar of support for the rouble.
Dmitry Polevoy at ING Bank said in a note that payments for the last large tax, profit tax, could act to check losses in the rouble later in the session.
Russian shares were mixed on Monday, with the discrepancy between the country's two major indexes reflecting moves in the rouble.
Russia's dollar-denominated RTS index was 0.2 percent weaker at 854 points, while its rouble-based peer MICEX traded 1.1 percent higher at 1,589 points.