04/01/2014 10:23 am ET Updated Apr 01, 2014

Everyone Seems To Be Wondering If Josh Elliott Will Be The Next Matt Lauer

Ray Tamarra via Getty Images

Now that Josh Elliott has left "Good Morning America" for NBC Sports, rumors are flying that the former morning show host could be the next Matt Lauer.

Elliott accepted an offer from NBC Sports over the weekend, after months of tough negotiations with ABC News. He was replaced by Amy Robach on "GMA," starting Monday.

Immediately, people started wondering if Elliott could eventually replace Lauer as co-host of "Today." Then, the New York Post and New York Daily News spoke to sources who said that Elliott is gunning for Matt Lauer and Bob Costas' jobs. That story was picked up by other outlets.

To be clear, Elliott's new role is with NBC Sports and a non-compete clause in his old contract prevents him from appearing on NBC News programs for six months. NBC News president Deborah Turness also insisted to the Post that Lauer is the man the network wants for the job, saying, "There is nobody I would rather have in the anchor chair than Matt.”

Lauer's contract is up in 2015, and the "Today" co-host — who has been the subject of negative press since Ann Curry was ousted from the show — has not had an easy time in his role in the past two years. NBC Universal News Group chairman Patricia Fili-Krushel said last year that the show will focus on finding Lauer's successor, and that she would "keep Matt Lauer as long as he wants to stay on the 'Today' show."