New Orleans Saints
The New Orleans Saints would almost certainly take a chance on James Washington, especially with the lingering Michael Thomas injury. The Saints will also have a new quarterback under center, after the retirement of Drew Brees. I am sure that whoever the new quarterback is, between Jameis Winston or Taysom Hill, they would love to have a young, liable receiver on the outside.
This would be an excellent opportunity for James Washington to shine, that being by complimenting Michael Thomas or filling the void for him if he does have to miss time. It would not be hard for him, to move up on the depth chart. He would immediately step into a tremendous role in this offense.
It should be noted that there have been more recent reports indicating that the trade request was false. Regardless if they are true or not, these are the best destinations that I feel would suit James Washington.
Washington is a young player, with lots of talent, that has not been unleashed for the rest of the league to see. It is safe to say that he has not even scratched the surface of his full potential.