Although the character he's taking to is not named, judging by the dialogue I'd hazard a guess that it's the younger version of Charles Xavier (James McCavoy), and this is part of their first exchange when Logan travels back to try to convince him to help.
The ultimate X-Men ensemble fights a war for the survival of the species across two time periods in X-Men: Days of Future Past. The beloved characters from the original X-Men film trilogy join forces with their younger selves from X-Men: First Class, in an epic battle that must change the past -- to save our future.
X-Men: Days of Future Past stars Hugh Jackman, James McAvoy, Jennifer Lawrence, Patrick Stewart, Michael Fassbender,, Ian McKellen, Nicholas Hoult, Omar Sy, Peter Dinklage, Halle Berry, Adan Canto, Fan Bingbing, Anna Paquin, Evan Peters, Daniel Cudmore, Ellen Page & Booboo Stewart. The movie will hit theaters May 23, 2014.