1989 is probably the best argument we had. 10-1 and SEC champs is pretty strong.
It would be tough. That season also had 11-0 Big 8 Champ Colorado, 10-1 independent Miami, 10-1 Big 10 Champ Michigan, and 11-1 independent Notre Dame. Plus 10-1 SEC Co-Champ Alabama, and 10-1 SWC Champ Arkansas (terribly weak schedule though).
-Michigan would seem to gets in on account of major conference champ and due to schedule strength that year (and that one loss was to Notre Dame).
-Colorado likely does as well between the combination of undefeated, major conference champion, and strong schedule (they beat an Illinois team that finished in the top 10, a ranked-at-years-end Washington, and top 5 Nebraska to win the conference).
-Notre Dames schedule was very strong (they beat: 10-3 ACC Champ and ranked-in-mid-teens UVA, the above Michigan team, a ranked Michigan State, 8-2-1 Pac 10 Champ and close to top 10 USC, then-top-10 Pitt (who would stay ranked all year but also lost another 2 more games to ranked teams), and a ranked Penn St. Their only loss was to top 10 Miami).
Itd have to be sorted out between 10-1 Miami (win v the ranked Pitt team mentioned above, the above Notre Dame, the ranked Michigan St team mentioned above, and loss came to highly ranked FSU), 10-1 Alabama (win v the mentioned Penn St + top 10 UT and lost to top 10 Auburn) and 10-1 UT (win v top 10 Auburn, 8-3 Duke, and lost to 10-1 Alabama... SEC was unfortunately pretty bad that year and the #6 UCLA we beat ended the year an unranked 3-win team).
In the end it would probably have to be either Miami or - if not - possibly Alabama in that last spot over the Vols.