are you talking about trumpet vine, campsis whatever it is?

It takes some killing. I cut one down six years ago when we moved in. Cut, pulled, and sometimes sprayed the stumps with round-up. This year I had only one tiny little shoot and it hasn't come back since I pulled it. So maybe next year, none. But it has taken this long. Each year was easier. They have extensive root systems and will shoot up anywhere there is a living root.

Tiki
There are several vines that are nicknamed Hummingbird vine. The most difficult one to get rid of is the trumpet vine (Campsis Radicans). If that is what you have then it may take several applications of Roundup.

Penny