Why not for NYI? They have said their piece by now. He will either re-sign or not. These next several days aren't going to change that. Why not get something for a diminishing asset?
If they are afraid the team they trade him to will sign him, that will pretty much happen regardless. Let's say they trade him to San Jose June 15th for 2 2nds, and SJ makes its pitch and he signs June 21st before the draft. They lose him and get 2 2nds for it. Terrible result. But let's say they don't trade him. San Jose is making the same pitch on June 24th or whenever the negotiation period opens. If the pitch is good, which nothing changed in the scenarios to change the pitch, then Tavares signs July 1. They get an extra 10 days or so, but not the 2 2nds. That is worse than the alternative. If they trade him, its still possible he listens to SJ, waits until July 1, listens to everyone else, and signs back with NYI, who is 2 2nds richer.