Edit: I just realized that you are asking about different time period. I can't vouch for slavery in Eastern Europe during Atlantic slave trade but I've written about slave trade during ~700-900 AD that impacted Eastern Europe and Middle East. I'll leave it for mods to decide.

Yes, there was a huge demand for slaves in Middle East from Slavic territories, modern day Eastern Europe/Russia. I don't have exact figures but the number of slaves captured/traded from Rus' and before it's formation was huge, compared to the beginning of Atlantic slave trade. Islamic states couldn't enslave those of the same faith, also they weren't supposed to enslave Christians and Jews because they were of the same Abrahamic religion. What was left for them is pagan Slavic tribes. It was a fair game and Khazars and Bulgars would sell captured slavs to the Middle Eastern states. The territory was called "Sclavia" and captured slaves were called "sclava", thus this is where we get the word from. Origin of word goes from Arabic to present day like this: Saqlab - Sclaveni - Sclavus - Slave. The region was so profitable in slave trade that this is where Vikings come in, called Varagians. Varagians were lured in to modern day Russian territory for the same lucrative trade of slaves and other commodities. Official Nestor's Chronicles about old Rus' state that Varagians were invited by the people of Rus' to govern them and keep order. It is likely that it's a "nice" version of how it was. Since historians still can't agree on the origins of Rus'. Harvard's academic Pritsak proposed that Rus' was not referring to ethnicity but it was actually an international corporation! Such an unusual proposal but he claims that Varagians were in competition with another trading company and Rus' (their company) initially was lured in to Volga trade route for the slave trade. Then Variagians broke off links and started their own political ruling class that became Kiev Rus'. Pritsak argued that after uniting more and more Slavic tribes and with expansion of their new found state, they had to cut down on slave trade and focus on other commodities because Varagians got eventually absorbed into the Rus' peoples. So in summary, yes there was a huge slave trade in Eastern Europe and the subject is pretty sensitive in Russia because there are some convincing arguments that the establishment of Rus' was more of a commercial enterprise that evolved into a clan ruling political system.

Sources: Origins of Rus by O. Pritsak, along with undergrad course on pre Peter the Great Russia.