The Crimson Quarry has an excellent post running down candidates in detail that sections of this post are derived from.

The Fringe

Schools that have been brought up at one time or another but are not worth a fuller discussion for various reasons.

Schools that would say no

Texas. Blame those Texas newspaper articles describing UT's flirtation with the Big Ten after the SWC exploded, but Texas comes up whenever this topic does. Despite the travel involved the Big Ten would do that in a heartbeat; Texas is a fantastic school that opens up copious television markets and is a national power in both football and basketball.

Texas would not, though. They are the master and commander of an entire conference with weak revenue sharing relative to the Big Ten. They have longstanding rivalries with virtually everyone in the Big 12 South. And their nationally competitive baseball program would be badly hurt by joining what's basically a mid-major conference.

Nebraska. Massive football tradition and geographically somewhat feasible but there's no way the Cornhuskers would give up a 100 years of rivalries for the Big Ten. Have no links to anyone in the conference.

Notre Dame. If the Big Ten is doing this when Notre Dame's NBC contract has six years to run, the Irish are not in the mix.

Schools that don't offer enough

Iowa State. Why on earth would anyone want Iowa State in their conference? No TV market and no success in either major sport. If Iowa doesn't want them, and I'm sure they don't, why would anyone else?

West Virginia. Tier III institution would probably get rejected by the presidents. Good programs in football and basketball but brings zero recruiting base and zero television market. If the only considerations were on-field performance they'd be the obvious #1 choice but all their peripherals are poor.

Cincinnati. Legitimate traditional basketball power (two national titles in the 60s to go with the Huggins era) and nouveau riche football school, but probably destined for a major drop with Brian Kelly out the door. Academically, a non-starter: it's a tier III commuter school.

Louisville. Geographically and athletically plausible but a tier III institution.

Prime Candidates

Rutgers. Hypothetically brings New Jersey and New York markets into play, except few really care about Rutgers when they're not good and they've rarely been good. Very rarely. Basketball program a nonentity; football was a nationwide punchline until the arrival of Greg Schiano, at which point they've had one standout year and a bunch of middling ones that end in nondescript bowl games.

Missouri. Geographically adjacent and has longstanding, if on-and-off, rivalry with Illinois. Good, not great, state school that would be the worst-ranked school in the league but not by much, especially after a post-CIC bump. Brings a new, large TV market into play. Also brings Don Draper with it.

Negatives: neither football or basketball is the sort of program that brings any wow factor, though the football program is a solid and developing one under Gary Pinkel. And Mizzou has been in the Big 8/12 since its inception. Fevered rivalry with Kansas and the sort of non-rivalry with Nebraska that saw Mizzou on the end of dozens of heinous beatings to the point where if Pinkel hadn't run up the score in the final year of Callahan he would have taken he would have taken heat for it.

Would Mizzou go? I mentioned them on the radio yesterday, at which point someone who grew up in the area called in to cast doubt on the possibility the Tigers would even consider leaving the Big 12. He certainly knows better than I do. On the other hand, some Mizzou folks have started a pro-Big Ten blog and the Rock M Nation thread discussing BHGP's discussion of a potential move is split down the middle. The local paper's Mizzou beatwriter, however, is adamant:

RT @Kevin_Baum What's your take on mizzou's chances of joining the big 10? ... To quote Dean Wormer, Zero Point Zero

I don't know. I expect that Mizzou would at least flirt with the Big Ten in an effort to get the Big 12's revenue sharing increased.

Pitt. Obvious natural rivalry with Penn State that makes the Nittany Lions less of an odd duck in the league both geographically and culturally. Brings another TV market, though Pittsburgh is an area that already gets the BTN. Rich tradition in football and has been intermittently decent over the last decade; basketball program has recently built itself into a national power but has little in the way of history.

Scholastically Pitt would be an average Big Ten team, which is very strong relative to other serious candidates. And there's no question whether they would jump or not: Pitt would kill to get in the Big Ten. They'd get to play Penn State, they'd get a ton more football revenue, the basketball would be fine, and they could play WVU out of conference.

Negatives: they play in a sterile NFL stadium that's usually half-empty, though a Big Ten fan with road-trip inclinations could view that as a positive. And adding Mizzou or Syracuse or whatever puts another state in the BTN footprint; Pittsburgh doesn't. And you could see this hurting Big Ten schools' Pittsburgh-area recruiting. Now players in the area can pick between the Big Ten or staying close to home; in the future they can have both.

Syracuse. Geographically somewhat awkward; football program has totally imploded since Paul Pasqualoni fell off. On the other hand, an excellent school (almost exactly on par with Pitt) with a powerhouse basketball program. Their location is a blessing and a curse: it's far away but also makes the Big Ten considerably more important in New York (state, not City).

Syracuse might like it in the Big East enough to shoot down an overture, though. They're decidedly more eastern than a lot of Big East schools.

Chart?

Grid! Grid of judgment!

A legend: teams are graded on a 3 point scale, where 0 is uncatchable a factor so poor it disqualifies the program in question, 1 is is an active detriment, 2 is "meh", and 3 is a positive. The "average" column does not include "willingness," since it's an attempt to judge the attractiveness of the teams only.

"Other sports" rankings derived from Director's Cup standings.

Your winners amongst the even distantly feasible: Pitt and Missouri, and Missouri is only distantly feasible. Both are clearly poor options relative to Notre Dame, but that ain't happening. Your projected 12th Big Ten school: Pitt.