They do have a nice catcher prospect of their own, named Jesus Montero, who they seem to be setting up as the next Thurman Munson. I'm not sure they'll be in on Mauer to do anything but muddy the waters. But I might be underestimating their willingness to spend money or overestimating their devotion to Montero (who, should the Yankees sign Mauer, would make one heck of a trade piece, just throwing that out there)
But there are also other well-financed teams out there that need catchers too. The Mets spring to mind quickly. If their owner isn't as badly off as the media was reporting, I could see him springing a lot of money for a franchise leader for his leaderless franchise.
I think our best bet for Mauer really is trade. there's too many crapshoot elements for me to believe we can safely trust to FA.