In Eragon, immediately after encountering the Ra'Zac near Dras-Leona, Eragon starts to travel to Gil'ead, taking wide margin of Dras-Leona. However, earlier in the book, it is mentioned that the King would be visiting Dras-Leona in a week's time. If I'm following the timeline correctly, and map correctly, it seems likely that Eragon was traveling perilously close to the King, almost directly on the route from Urû'baen to Dras-Leona, and that the King would have been able to spot Saphira. Is my logic flawed, or is this the case?
It seems most likely to me that Eragon would have been aware of it to some degree. Here's what I think happened.
Eragon was injured, some distance south of Dras-Leona. It seems likely he would remain far to the south for some time. The distance seems to be fairly significant from evidence seen in book 3.
Namely, Eragon is able to run (Albeit when tired) carrying a person over the course of maybe 6-8 hours south of Helsgrind to the spot where Brom was killed, or at least, reasonably close. I would guess that puts the distance at around 20-30 miles south of Helsgrind, already 20 miles south of Dras-Leona. That should be far enough away to avoid attention from the King.
His travel was very slow at first, as he didn't have much strength.
- It seems likely that the King didn't spend much time in Dras-Leona, only doing what he had to.
Given all of that, I think Eragon traveled near the spring of the Ramr river, staying far enough south so that he would avoid the King, and be safe.
Eragon did pass nearby, but we never learn exactly how close. Also, during this time Saphira was spending most of her time hiding. She was mostly in range of Eragon during that time, but it's possible she was far enough away from the King to remain unnoticed.
Also, it's equally possible that the King didn't follow his travel plans as advertised. He's very sneaky. :)