In a statement, the foreign ministry called on the government “and all armed groups who oppose it” to agree to ceasefires “without delay”.
Russia has twice vetoed Western-backed draft resolutions on Syria at the UN Security Council.
Its call came hours after clashes erupted in part of the Syrian capital.
The head of the ICRC, Jakob Kellenberger, had travelled to Moscow to discuss a ceasefire arrangement with Foreign Minister Sergei Lavrov.
For more on this story, visit: BBC News – Russia joins Red Cross call for daily truce in Syria.