The High Negotiations Committee (HNC), the main opposition group at the UN-sponsored talks, said it expects to meet a Russian delegation Tuesday in the Swiss city.
"We sincerely hope to persuade Russia to stand by the Syrian people," said HNC spokesman Nasr al-Hariri after his latest talks with UN Syria envoy Staffan de Mistura.
"We hope that Russia will bet on the Syrian people and will not bet on one person who has decided to destroy the whole of Syria so as to remain in power," he added, referring to Assad.
Moscow and Ankara are co-sponsors of a fragile ceasefire, brokered at the end of December. In Moscow, a foreign ministry source confirmed that deputy Russian foreign minister Gennady Gatilov would meet opposition negotiators along with all sides in Geneva.
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Gatilov and Sergei Vershinin, head of the ministry's Middle East section, are both in Geneva for the 34th session of the Human Rights Council, currently under way.
"In the framework of the intra-Syrian talks in Geneva we have meetings with all parties, including the opposition which is represented by more than one group," said a ministry source.
In Geneva, al-Hariri said he believed Moscow was showing signs of openness.
"It has tried to be neutral ... We see an openness in their position, especially vis-a-vis the revolutionary moderate factions," he said, referring to moderate rebels.
"The fact that it has recognised these factions as a negotiating party is an indication of Russia's openness and we are hopeful that tomorrow's meeting is also a positive indication.