To teach the Greek language, or writing, or philosophy, or math/science?
Quick answer is yes they did. Educated Greek slaves were very expensive and basically required for wealthy Romans. Even someone as austere as Cato had a Greek Slave to teach his Children the Greek Language. If you were wealthy you would have a very intelligent greek slave to teach your children or to be a sounding board for your own arguments. An example of the latter would be Diodotus the Stoic who according to Cicero was lkike the following "The Stoic Diodotus, another man who lost his sight, lived for many years in my house. It seems hard to believe, but after he became blind he devoted himself more strenuously to philosophy than he ever had before. He also played the lyre, like a Pythagorean, and had books read to him day and night; he had no need of eyes to get on with his work. He also did something which seems scarcely credible for a man who could not see: he continued giving lectures on geometry, giving his pupils verbal indications of the points where they should begin and end the lines they had to draw"
Now for less wealthy Romans a Greek slave was still wanted so they would buy someone who had knowledge of the language and if possible the literature of Greece to act as a teacher to their children.
Most of the teachers mentioned in various Roman texts can be seen to be Greek another example of the greek schoolteacher is a slave that was brought by Catulus then latter given his freedom and a large sum of money to set up a school.
If you want to read a little more this book could help you it is where i got this info from, even though it is rather old. But in been so old it is now free.