Unfortunately, no. If verdigris is present but hidden inside the tube, there is not much a homebrewer can do to remove it. Never use acetic acid (vinegar) to clean your chiller, as this seems to be the fastest (but not the only way) for verdigris to form.
I would stick with wort on the outside of the tube. Trust me, it's not any faster, and is more labor intensive.
To expand further, the fixation on acetic acid comes from formation of copper acetate, one form of verdigris, if the acetic acid is allowed to remain on the copper when it dries. Oxidation then forms copper acetate, rather than the more benign copper oxide or carbonate. Copper acetate is very dark, almost black, and is easily distinguished from the much lighter, faintly green oxide coating. Definitely take special care to rinse off all the acid, whether from the wort or vinegar cleaning solution before storing.
Since the wort itself is a fairly good copper cleaner, I have to disagree with Redneck's admonition to not use vinegar to clean the chiller coil. If the oxide layer is left on the coil, and the coil comes out shiny clean from your wort, you can be reasonably certain the copper oxide was left behind in your brew. In fact, in clear cleaning solutions, you would see the dark colored nodules that precipitate out of solution.
Vinegar alone, or especially with a pinch of simple table salt in solution, is one of the best, most readily available cleaners for copper. The brass used in pipe fittings is an alloy of copper, and also is cleaned in that same solution. Again, be sure to rinse off all the acid before storing, and make double sure the little nodules of (possibly verdigris) precipitate are flushed away.
There is something to be said for fastidious attention to cleanliness and possible toxic contamination in your food processing. I haven't gone the extreme route of all stainless, all welded tubing with tri-clover fittings, but my consumption is relatively low. For what it's worth, it is said that yeasts feed on the minerals and metals, and may find minute quantities healthful.
If you scour the web long enough and take all the admonitions literally, you'll eventually conclude that no substance is safe for use or consumption. You should make an informed decision of what's reasonable. Do some experiments. Copper and vinegar reactions are commonly taught as early, relatively safe experiments for young, budding chemists. You'll find lots of teaching material on the web. See for yourself what's possible, but keep in mind the concentrations you'll encounter in your brewing.