When the Big10 a few years back provided an offer to ND and got snubbed, they basically said, "OK, we offered, that's it, we stay as we are". The logic here is "we will stay at 11 unless Notre Dame is added". That in itself is enabling Notre Dame independence and making Notre Dame appear so special and beyond reach.
If the Big 10 wants to be 12, that, in itself, needs to be clarified. Then, they set out out to find their "top choice among prospects willing to be considered", rather than all or nothing on only one school.
When the SEC was expanding to twelve, they said so, and commenced interviewing schools, first luring Arkansas and after talking to FSU and Miami, settled on South Carolina who lobbied hard to be picked. When FSU expressed a desire for the ACC, the SEC did not stop looking for #12, they focused on other viable options.
When the ACC recently expanded, they got to 11 via compromise, with one school they most wanted (Miami) and another (VPI) who was not initially desired. They kept pressing for 12, wanting that Championship game, and brought sought after BC on-board. It was messy, but they got what they wanted. Syracuse got left out.
The point is, if the Big 10 decides for 12 again, make an offer to Notre Dame if that is their first choice, and if they decline, have a working list in rank/order of preference be it Mizzu, Pitt, 'Cuse, ISU, Nebraska, or whomever.
While it is possible a certain school in the current Big12 schools may not seek to switch to the Big10 as some may think, there are a few in the Big East and elsewhere who would. It is possible the Big 10's second choice would not switch. One looks till they find an agreeable match. The Big10 certainly has more desirable options, numerically, than the PAC 10. No pick is going to be perfect, perhaps Notre Dame included.