The Ring now has passed beyond my help, or the help of any of the Company that set out from Rivendell. Very nearly it was revealed to the Enemy, but it escaped. I had some part in that: for I sat in a high place, and I strove with the Dark Tower; and the Shadow passed. Then I was weary, very weary; and I walked long in dark thought.'
The Two Towers : The White Rider
What did Gandalf mean when he said he 'strove with the Dark Tower'? Was this possibly a battle of wills?
It's implied that this was a battle of wills, yes. It's useful here to look at how it played out from Frodo's perspective at this time (from The Breaking of the Fellowship):
And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir - he threw himself from the seat, crouching, covering his head with his grey hood.
He heard himself crying out: Never, never! Or was it: Verily I come, I come to you? He could not tell. Then as a flash from some other point of power there came to his mind another thought: Take it off! Take it off! Fool, take it off! Take off the Ring!
That this event is explicitly linked with Gandalf's statement is confirmed by the following from the chapter The Black Gate is Closed:
Yet even as he spoke his last words to Saruman, and the palantir crashed in fire upon the steps of Orthanc, his thought was ever upon Frodo and Samwise, over the long leagues his mind sought for them in hope and pity.
Maybe Frodo felt it, not knowing it, as he had upon Amon Hen, even though he believed that Gandalf was gone, gone for ever into the shadow in Moria far away...
And so we can say with absolute certainty that the other thought instructing Frodo to take off the Ring was definitely Gandalf's.
Gandalf: On a hill in Fangorn wrestles in thought with the Eye of Mordor, and saves Frodo from yielding.
The "high place" Gandalf says he sat on was a hill in Fangorn, and when he says "strove with the dark tower" he means he "wrestled in thought with the Eye of Mordor".
This was also hinted at it in the book with the third "Fool, take it off!" voice that Frodo hears on Amon Hen, but here is a direct confirmation from Tolkien as to what happened.
This part of Tolkien's scheme was first quoted in J.R.R. Tolkien: A Reader's Companion, and a facsimile of the page was later published in Maker of Middle-earth and Voyage en Terre du Milieu. For everything we know of the entire time-scheme, see this answer.
