If your project is public, please share it with question in future, it makes much easier to know the context of your problem.
Have you checked your texture, making sure it has no such line at the edge?
Sometimes when you resize images in Photoshop with non-background layer, it will slightly blur edges, and then if you save it as JPEG or any other non-transparent format, it will put white colour under.
That is one option.
Other is that you might have texture that has only some part in it that you refer in UV channel. Due to texture filtering GPU will pick texels from around to make sure texture looks not pixelated and is nicely filtered on different angles, including on distance where it uses lower resolution (mip) textures automatically.
So if your UV is referring only to some area in texture, make sure you add extra padding around pixels - this is general practice when authoring textures.