No, it is not possible to create a buffer with a variable width directly on a line feature. However, it is possible to achieve a similar output to a buffer with a variable width by using representations.
Create a representation on a feature class and modify the properties of the representation rule for individual features. For example, use Tapered Polygon as the geometric effect to create a dynamic polygon along a line feature whose width varies by two specified amounts along its length. For more information on tapered polygon and other geometric effects, refer to the following ArcGIS Web Help page: List of representation geometric effects.
A representation can only be created on a feature class. Creating, modifying, or deleting representations
requires an ArcGIS for Desktop Standard or Advanced license. Representations can be viewed and drawn with
an ArcGIS for Desktop Basic license.
For more information on how to work with representations, refer to the following ArcGIS Web Help page: A quick tour of representations.
Users can also construct a Python script to achieve such output. However, this requires some Python knowledge.