[Scharlemann and Thompson][1] proved that one can isotope/slide a tunnel
to be disjoint from a minimal genus Seifert surface (which was what I had in mind
in my comment above). Their argument does not depend on it being a knot complement.
I also found a [citation for a paper of Sakuma][2] which claims to prove the
result for punctured torus bundles, so it might have the result you're looking
for (although I haven't looked at the article yet). 


  [1]: http://www.ams.org/mathscinet-getitem?mr=1990938
  [2]: http://www.ams.org/mathscinet-getitem?mr=1483404