The wiki claims- unfortunately without attribution- that Dumbledore offered to be the Potters' Keeper, and was turned down.

I definitely remember this from the third book. The adults are talking about the Potters' deaths in the Three Broomsticks Inn and someone mentions that Dumbledore himself offered to become the secret keeper, but was turned down with insistences that Sirius Black would never betray them.

EDIT: Found it.

"So Black was the Potters' Secret-Keeper?" whispered Madam Rosmerta.
"Naturally," said Professor McGonagall. "James Potter told Dumbledore that Black would die rather than tell where they were, that Black was planning to go into hiding himself... and yet, Dumbledore remained worried. I remember him offering to be the Potters' Secret-Keeper himself."