"President Talabani has come back from a coma," Fars cited sources close to the president's medical team as saying.
Officials in Baghdad, however, did not initially confirm the report, Fars said.
Talabani, an iconic figure representing Iraq's Kurdish population in the government, was taken to a Baghdad hospital after his stroke and was flown to Germany for further medical treatment a few days later.
Iraq's presidential office denied media reports that Talabani had died in a Baghdad hospital a week after the stroke.
Meantime, Talabani's wife Hiro Ahmed Ibrahim and a delegation from Talabani's Patriotic Union of Kurdistan (PUK) party have reached Tehran for bilateral talks with Iranian officials, Fars said.
Talabani's recovery will play a key role in soothing ongoing ethnic and sectarian unrest in Iraq, according to observers.