Everything below is my subjective opinion:
One issue you have with swapping out op amps on a card like that is you don't necessarily know what the circuitry is like - where gain is involved rather than just a buffer there might be compensation capacitors in the feedback loop that do bad things if you use an op amp that isn't unity gain stable, for op amps that aren't internally compensated they might not be stable if buffer stages and/or require a higher gain than the card is setup to use and then you have some complications with the power supply - whether there is enough voltage and whether it is setup for single or dual supply support, etc.
The op amps that tend to be fairly forgiving and work in most cases would be stuff like the OPA2132/2134, JRC4556, LM4562, the NE5000 series, etc.
The NE5532 is a "workhorse" amp largely transparent but a bit dated now and can sound a bit "raw".
A lot of people like the OPA2134 as its largely transparent with a fairly mellow sound but personally I think it is like someone playing the same tune for the 1000th time - accurate enough but with no passion - I prefer the OPA2227 as a replacement as it sounds a bit more "involved" with the music. (Or even better OPA2228 but you'd need carnal knowledge of the circuitry to know that would work in there).
LM4562 is very detailed, crisp and punchy and in buffer positions often allows the good bits of any previous gain stage through as well i.e. if the op amp in the gain stage has good bass the LM4562 tends to leave it relatively unmolested. However it can sound a little robotic and power supply noise can result in it being a bit metallic and harsh. One of my favourites however. I'm not sure what is going on with regards to manufacturing though as TI did say they were ending production of it in DIP format but most places still had good supply last I checked.
NJM2114 generally has very good stats but IIRC the bass is a bit lean and the treble a bit fatiguing.
One thing that might be worth trying is the LM4562 or one the LME equivalents with a 10uf "boutique" capacitor over its power supply pins i.e. Nichicon FG, KZ or Elna Silmic II in my experience that helps to give the low end more satisfying body to it and a cleaner background.
One of my favourites for bass is the AD8066 paired with LM4562 buffer(s) but you'd need to know the circuit supported it as it isn't just a drop in replacement (neither is it available in DIP format).
If you feel like playing around with adapters then the AD8610 in the IV stage might be worth a consideration.
I suspect the circuitry is very similar to fig 36 on page 35
http://www.ti.com/lit/ds/symlink/pcm1792a.pdf so the transimpedance op amps would probably have to be unity gain stable to avoid complications and destroying sound quality.
EDIT: I usually buy my op amps from Farnell or Mouser (pretty much guaranteed you won't get fakes) but Mouser ships from the US so is a bit expensive for small value orders.
