Kelly probably was not the top candidate on Notre Dame’s wish list when the process began, but as more than a week has passed since the firing of Charlie Weis, he has emerged at the front of the line. Kelly has said he is open to the job, he fits Notre Dame’s Irish Catholic archetype and he would be able to compete for the national title game much easier at Notre Dame than at Cincinnati.

Image Cincinnati Coach Brian Kelly is expected to interview with Notre Dame on Tuesday. Credit... Keith Srakocic/Associated Press

A Cincinnati spokesman said that the Bearcats were negotiating a new contract with Kelly. But Kelly has not hid his interest in Notre Dame. On Monday he wrote on Twitter: “Just informed our team that Notre Dame has contacted me and I will listen to what they have to say.” (His Twitter followers jumped to near 3,500 from around 1,800 in about four hours.)

Kelly has moved to the front of the line because high-profile candidates have publicly removed themselves from the running  Oklahoma’s Bob Stoops, Florida’s Urban Meyer, Iowa’s Kirk Ferentz and Northwestern’s Pat Fitzgerald, to name a few.

Swarbrick has said he wants a proven winner at the college level, which essentially eliminates any of the dark horse N.F.L. candidates like Jon Gruden and Tony Dungy.

Stanford Athletic Director Bob Bowlsby said Monday he was “100 percent confident” that his coach, Jim Harbaugh, was not a candidate. He said there was “absolutely no chance” of Harbaugh’s being the next coach at Notre Dame, as he has been speaking with Swarbrick, his close friend, throughout the process to give him advice.

If Kelly says no or Notre Dame does not want him, the pickings appear slim. The New York Post reported Randy Edsall, the Connecticut coach, as an emerging candidate. A Connecticut official said it was the university’s policy not to speak about other programs’ coaching searches.

Edsall has done a great job of building the UConn program from scratch, but his résumé pales compared with Kelly’s. In the past two years in the Big East, Kelly has gone 13-1 while Edsall has gone 6-8.