Now, that is a possible scenario, where saying he wants to play OF might actually be a clue to other GMs that he does not want to sign with the Sox as a DH. My guess is, then, that he might be willing to take less money, and that's why they want to bring more teams into the picture.
JD can easily get 30-40 games in LF with the Sox without any injuries to anyone. Maybe, that's not enough.