I've checked this question out, and Keen offers a great explanation for the seeming plot hole, but my question is a little different.
After reading this reddit post, I'm really curious whether evidence/hints of this secret plan appear in either the books of the films.
The Secret Plan
I've summarised the theory below, see the link above for the unabridged version
Gandalf secretly planned on taking the fellowship to where the eagles live and having the eagles fly them to Mordor. The eagles lived on the other side of the Misty Mountains but all the routes for crossing them were too dangerous and difficult, and Gandalf (along with his secret plan) ends up falling down a chasm in a battle with the Balrog. Just before falling with the Balrog he tries to surreptitiously tell them the secret plan but was too subtle and they didn't understand. When he came back as Gandalf the White he had forgotten many things, including the plan to meet the eagles.
The most alluring evidence is Gandalf the Grey's last words:
Fly you fools!
which could possible be interpreted as a hint to the Fellowship (necessary because Sauron has ears everywhere and preserving the element of surprise is vital)