Yes it is possible, it takes at most 3 days and it's free of charge.
You need to contact the network operator you are moving to and apply for number portability. You can contact them online, by phone or you can visit any Germanos or Cosmote shop.
If you are living/registered in Greece you will need to bring your ID Card and your VAT identification number. If you aren't, I guess that they'll only need your passport.
Detailed information here (click on the 'Join Cosmote Network'):
COSMOTE: Support