DL offers this directly in the app or on DL.com (if available) -- they don't wait until DOD as AA/UA do
FWIW, I have seen it offered for TATL but it's random (it was never offered to me on JFK - MAD last month despite D1 going out with a couple of seats empty, on the flip side I was offered a paid upgrade on my JFK - LHR award ticket in July and last year had a couple TATL flights with an offer around $900).
FWIW, based on availability per EF I doubt you'll get an offer -- DL won't oversell D1 and sitting at J2/Z1 for 6/12 I'm guessing they'll hold any upgrades for the gate for walk-up fares or IRROPs (which means no cash upgrade). For 6/26 it's not looking great either with Z0 15 days out. Plus neither flight looks to be in danger of an oversell in Y so unlikely to be any op-ups.
6/12:
J2 C2 D2 I2 Z1 W2 Y9 B9 M9 H9 Q9 K9 L9 U9 T9 X9 V9 E9
6/26:
J3 C3 D3 I3 Z0 W0 Y9 B9 M9 H9 Q9 K9 L9 U6 T3 X3 V0 E9