The Notre Dame series takes a hiatus on Michigan's 2018 and 2019 schedules. I am assuming that Michigan would seek to replace with the Irish with an opponent of comparable stature.
(Feel free to insert your favorite joke about how Notre Dame really doesn't have the stature everyone assumes. The fact is, the media treats Notre Dame as if they were an elite team, whether they are or not.)
I checked the future schedules on fbschedules.com, of every opponent that could be considered a plausible replacement for Notre Dame. I even checked a few that are marginal, like Syracuse and Pitt. But I didn't check teams like Duke or Louisville that no rational person would consider a like-for-like trade with Notre Dame.
I assumed that no school would agree to schedule Michigan if they already had a tough out-of-conference game scheduled in either or both of those years. For instance, Oklahoma and LSU have a home-and-home scheduled in 2018-19. Scratch both of them off the list. (One of the premium message boards reported this week that Dave Brandon had approached Oklahoma, and was told no thanks, though it didn't say for what years.)
These are the findings:
I. Definitely Unavailable
Oklahoma and LSU (home & home vs. each other)
Texas A&M and Oregon (home & home vs. each other)
Texas (USC, @Maryland in 2018; Notre Dame in 2019)
USC (@Texas in 2018, in addition to the usual Notre Dame h/h)
II. Probably Unavailable
Alabama (hosting Georgia Tech in 2019)
Georgia Tech (hosting Ole Miss in 2018; @Alabama in 2019; plus Georgia h/h)
Stanford (Virginia in 2018; @Northwestern in 2019; plus Notre Dame h/h)
III. Possibly Available
Florida and Florida State (home & home assumed; nothing else scheduled)
Miami (only thing scheduled is Rutgers home & home)
Georgia (only thing scheduled is recurring home & home vs. Ga. Tech)
Syracuse (only thing scheduled is Northwestern in 2019)
IV. No Known Conflicts
The open question is whether Dave Brandon and Brady Hoke want to play a September game in an outdoor stadium in the south, where the weather conditions would be in the opponents' favor. If you take the southern teams out of the picture, you're left with teams like Pitt and Syracuse, which many fans would view as being less interesting opponents than Notre Dame.
It is a myth that the SEC schools never play tough out-of-conference opponents. Besides Alabama and LSU (already mentioned above), Tennessee has future home & homes vs. Oklahoma and Nebraska; and Arkansas has both TCU and Texas scheduled.
On the other hand, Georgia never seems to schedule a tough out-of-conference opponent, other than their annual tilt with Georgia Tech. Likewise, Florida seems only to schedule Florida State and occasionally Miami. Auburn is rather timid: a trip to Kansas State is the toughest future game they've scheduled.
Bear in mind that many of these schools could have conflicts due to ongoing negotiations that have not yet become public. And some of them just might not want to play Michigan. So there might be only one or two real options that would strike the average fan as an upgrade over just playing Notre Dame those two years.