You should be able to get TSA-Pre with Delta (as green card holder) as long as your Known Traveler number is on your Delta profile.
I just came back from a trip with my parents who are green card holders, and they both got TSA pre-check both ways.
We all got TSA-Pre check by getting Global Entry.
Last edited by lookcle; Dec 3, 2013 at 1:40 pm
Reason: Adding more