The Curator

The curator of the National Gallery in London was a mysterious individual who had insight into a particular Time Lord painting. He gave a coloured scarf (similar to the fourth Doctor's one) to Osgood.

After the Eleventh Doctor remembered his face, the curator told the Doctor, "in years to come, you might find yourself… revisiting a few. But just the old favourites, eh?"

He spoke with the Eleventh Doctor and told him that the name of the picture was in fact Gallifrey Falls No More, and pointed the Doctor in the direction of a search for Gallifrey, telling him he had "a lot to do."

The man then claimed to have perhaps been the Doctor once, or that the Eleventh Doctor was once him. He agreed with the Eleventh Doctor's suggestion that the Doctor would one day become the curator. (TV: The Day of the Doctor)

Behind the scenes
The script never directly states that this character is an incarnation of the Doctor, but it certainly implies that he may be a future version of the Doctor who has reassumed the form of his fourth incarnation, or in fact, a previous incarnation. Further fueling the possibility is that Tom Baker is not separately credited for this performance, receiving only a credit for playing "The Doctor".