At 0728 GMT, the rouble was 0.2 percent weaker against the dollar at 62.38, little changed from June's closing level of 62.77.
Against the euro, the rouble shed 0.4 percent to 73.15 compared with 73.29 at end-June.
In the past few days, the rouble was supported by the tax payments, which ended on Monday.
It may come under pressure in coming days as foreign investors in Russian companies convert rouble-denominated dividends.
But Otkritie brokerage said it expected the rouble to move towards 62 per dollar as emerging market currencies looked oversold.
Russian stock indexes inched lower as prices of oil, Russia's key exports, pared earlier gains. Brent crude futures were down 0.36 percent at $74.70 per barrel.
The dollar-denominated RTS index declined 0.2 percent to 1,160.1 points, while the rouble-based MOEX Russian index was 0.05 percent lower at 2,294.5 points.