I'm not sure what you mean by root suckers. Clematis only send up more shoots from around the base, the more the shoots the better. Some clematis will reseed although I've haven't seen that with viticellas. Could your root suckers be from other plants? Are you cutting your viticellas back in the winter?

Don't use Round Up. Whole clematis have been killed by people that accidentally sprayed too near the plant.

Katie

I agree with Katie: you can possibly have multiple shoots from the crown (which is what we all want with our Clematis), but not root suckers.
What makes you think you have viticellas?

Katie, thanks for your reply!

I do cut them back late winter. I just assumed it was root suckers as they are appearing quite far from the base, around a foot or so. It is definitely more viticella species and not some other plant, I saw them bloom last summer.

I guess if it's my viticella that's just thriving, sending up lots of shoots and being super-happy, I should be so as well, but it's actually a bit too much right there and it's shooting in the wrong direction - there's no trellis where it's going...

You can gently train it to climb the trallis. No harm to the adjacent plants - the rootball of your Clematis does not spread.

Janni, are you saying that there is a foot of space between the vines of the main plant and the shoots that seem like suckers?

These are not root suckers like you get on fruit trees, these are just more vines and they are a good thing. Nornally.

I did get rid of a tangutica that was sending them up two feet in all directions and not growing tall where I wanted it to grow.