Is there a way to upload an asset with a URL?
Firstly, you could manually copy/paste all your images into your Asset Source folders, then hit "Update Asset Indexes" from the settings. What this does is scan the directories for files, and creates them in the database if they don't exist. Once thats done, in your entries, you simply need to enter the filename in the search field (once you click "Add an Asset" on your asset field), and you're done.

For a more automated solution, you could look at Feed Me to update your entries, along with automatically uploading your assets from a provided URL. Your feed could be as simple as:

[{
  title: "Entry Title 1"
  images: [
    "http://my.website/my-image-1.jpg",
    "http://my.website/my-image-2.jpg"
  ]
}, {
  title: "Entry Title 2"
  images: [
    "http://my.website/my-image-3.jpg",
    "http://my.website/my-image-4.jpg"
  ]
}]