ORD-DEL is a no-brainer .. take the AA flight.

If you need to go elsehwere within India .. as CPX mentions Air Sahara.

Certain days of the week CX has flights DEL-HKG then a quick connection HKG-SIN will work.

Back to ORD from SIN you will probably need to do CX back to HKG and then to the US .. or SIN-HKG-NRT-ORD using CX and AA.

Look into RTW ticketing on this little junket! Tho unsure if miles to upgrade will work on a RTW.. At least you can do OneWorld all the way.