This statement was made in Hatcher's notes on 3-manifolds. It is assumed that these balls are smoothly embedded, and the theorem gives a smooth isotopy.
I believe this claim, it seems very clear to me, but how does one go about proving it? By induction? I can prove the statement for smoothly embedded balls in $\mathbb{R}^1$.
I was thinking that maybe we could emulate the proof of Alexander's theorem and get a morse function, take regular slices (which would contain balls in the dimension lower, which the inductive hypothesis would isotope to look very nice) but that's as far as I got.