In his timestream, he obviously did notice her long before "Asylum of the Daleks" -- before "An Unearthly Child", in fact. The First Doctor spoke with her when he was about to steal a TARDIS & to leave Gallifrey with Susan, as we saw in "The Name of the Doctor". However, it was only after "Asylum..." that he knew enough to connect one encounter between them with another (at the end of "The Snowmen"). As Clara herself said, he "hardly ever" (not "never") noticed her, most of the encounters were brief & (from the Doctor's point of view) decades or centuries apart, on different planets.
In addition, she was interfering with his timestream (albeit in a good way) &, as pointed out in "The Day of the Doctor", his timestream tends to protect itself from paradoxes by ensuring that things that shouldn't be remembered aren't remembered.