This is from the National Center for Home Food

Preservation:

How do I can oil with herbs? Can I can pesto?

Herbs and oils are both low-acid and together could support the growth of the disease-causing Clostridium botulinum bacteria. Oils may be flavored with herbs if they are made up for fresh use, stored in the refrigerator and used within 2 to 3 days. There are no canning recommendations. Fresh herbs must be washed well and dried completely before storing in the oil. The very best sanitation and personal hygiene practices must be used. Pesto is an uncooked seasoning mixture of herbs, usually including fresh basil, and some oil. It may be frozen for long term storage; there are no home canning recommendations.

There are only a few recipes that are safe for home canning that contain oil, and they are very highly acidified. Using oil in recipes not tested for safety can lead to botulism.