Barazan Sheik Ottoman, head of the presidential media office, told The New York Times by telephone Talabani was accompanied by several doctors after they established he was stable enough to travel.
Talabani, 79, was reported in stable condition.
Iraq's Foreign Ministry in Berlin confirmed Talabani was in the country, but didn't say where he was being treated nor provide details about his condition.
Talabani, a Kurdish politician who has been influential in mediating disputes among the country's political factions despite the limited powers of the Iraqi presidency, was rushed to a hospital Monday after suffering the stroke.
In an update Wednesday, the Iraqi medical staff at Baghdad Medical City said his "health is stable."
One of the country's two vice presidents, Khudayr al-Khuzai, a Shiite, has assumed Talabani's duties.