SANA news agency did not give details on the talks in Damascus, which come after recent meetings Bogdanov had with Syrian opposition groups in Lebanon and in Turkey.
"In Istanbul, I met with Syrian opposition members," Bogdanov said as he arrived at his Damascus hotel.
"We want to reach an agreement on many issues with the Damascus authorities," he added.
Bogdanov's Syrian counterpart, Faisal Muqdad, said: "I am sure our Russian friends are bringing with them ideas that we will discuss, so that the bid succeeds, and so that we can reach a political solution that is mainly founded on the integrity of Syrian territory."
No date has yet been set for any future talks.