Hi, 

Could you provide a photo of the area? It may help us to see the situation. 

Maybe you can pull or cut back the vine until it is just clear of the asparagus patch and carefully apply a systemic herbicide just to those cuts. Glyphosate will bind to the soil in the area where it dissolves into the roots, so potentially it wouldn't travel to the asparagus. Triclopyr or other systemic herbicides tend to travel through water or are more water soluble so they could potentially have more harm or uptake in the asparagus roots. 

Just be sure to read label directions and see what the half life is for each chemical listed. You can check at the National Pesticide Information Council. (this link is a general fact page for glyphosate - but you can search for others on this site) Waiting until after harvest would probably be best to be safe, or just pulling what you can until then. 

If a systemic herbicide were to get onto the asparagus it would probably end up killing those plants or damaging them at least. 

Feel free to send back a few photos and we may be able to give you more specific guidance. 

Emily

you're welcome