Sergei Lavrov, the Russian foreign minister, said: "We will be ready to support a resolution which will strengthen the demands of the International Atomic Energy Agency with regards to Iran and will ask it for an answer after a particular period."
The draft under consideration in the UN would make it mandatory for Iran to suspend enrichment and includes threats of sanctions if it does not comply.
Lavrov did not say whether Russia would back imposing sanctions if Iran failed to comply, striking a more conciliatory note than the US.
Vitaly Churkin, a Russian ambassador, said: "We are not in a rush at all. We do not want to ambush Iran in any way. We're very much in a negotiating political mode. We do not want to dictate things to Iran.
"Nobody's pushing, nobody's pushing Iran anywhere."
The resoultion comes after Tehran did not respond to a package of commercial and technical incentives that was submitted last month by six world powers in an attempt to persuade it to stop its uranium enrichment programme.
Foreign ministers of the six member countries - Britain, China, France, Germany, Russia and the US - met on July 12 and expressed disappointment that Iran had failed to respond positively to the package.
"A lot of time has passed, a lot more than the Iranian president promised our president in terms of when Iran would give its reply," Lavrov said in an interview published in Kommersant newspaper.
Russia is one of the five powers with a veto on the Security Council.