With meetings between Iran and world powers to resume in Geneva on Wednesday, Russia's deputy foreign minister Sergei Ryabkov was visiting Tehran, with both sides seeking a breakthrough.
"A more active Russian role is an important element in accelerating the final settlement of questions for a global nuclear agreement," Zarif was quoted as saying on state media.
With a final deal at stake by a June 30 deadline, the talks have stalled on key issues.
Ryabkov, who heads up the Russian negotiating team under the P5+1 talks, said good ties between Tehran and Moscow can still help "a rapid settlement of nuclear relations relating to Iran".
Since an interim accord struck in November 2013, two deadlines for a final deal between Iran and the P5+1 powers (Britain, China, France, Russia and the United States plus Germany) have been missed.
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Under the interim deal, Iran's stock of fissile material has been diluted from 20 percent enriched uranium to five percent in exchange for Tehran receiving limited sanctions relief.
Experts say such technical steps push back the "breakout capacity" to make an atomic weapon, while Iran denies seeking a nuclear bomb.
Iran's atomic agency chief Ali Akbar Salehi, however, said Sunday that Tehran was insistent on increased uranium enrichment, saying that in eight years it would need to produce 12 times more than at present.
Iran says it needs more enrichment to create fuel for peaceful nuclear energy production, but the issue has been a stumbling block in talks with world powers, who say a lower capability would suffice.
In Geneva, US Secretary of State John Kerry will meet Zarif on Wednesday, with lower-level P5+1 meetings on January 18.
The other main obstacle to a final deal is the timeline for lifting sanctions imposed on Iran for pursuing its nuclear programme in the face of international pressure.
Although Iran is led in the talks by Zarif and a team of negotiators, Ayatollah Ali Khamenei, as the Islamic republic's supreme leader, has the final say on any decision.